On full asymptotics of analytic torsions for compact locally symmetric orbifolds

We consider a certain sequence of flat vector bundles on a compact locally symmetric orbifold, and we evaluate explicitly the associated asymptotic Ray-Singer real analytic torsion. The basic idea is to computing the heat trace via Selberg's trace formula, so that a key point in this paper is to evaluate the orbital integrals associated with nontrivial elliptic elements. For that purpose, we deduce a geometric localization formula, so that we can rewrite an elliptic orbital integral as a sum of certain identity orbital integrals associated with the centralizer of that elliptic element. The explicit geometric formula of Bismut for semisimple orbital integrals plays an essential role in these computations.


Introduction
Let G be a connected linear reductive Lie group, and let θ ∈ Aut(G) be a Cartan involution. Let K ⊂ G be the fixed point set of θ, which is a maximal compact subgroup of G. Put Then X is a symmetric space, which is simply-connected with nonpositive sectional curvature. For convenience, we also assume that G has a compact center, then X is of noncompact type. Note that G acts on X isometrically and transitively. If Γ ⊂ G is a cocompact discrete subgroup, set Then Z is a compact locally symmetric space. In general, Z is an orbifold, where the singular strata of Z are indexed by nontrivial elliptic conjugacy classes of Γ.
Since G has compact center, then the compact form U of G exists and is a connected compact linear Lie group. If (E, ρ E , h E ) is a unitary (analytic) representation of U , then it extends uniquely to a representation of G by unitary trick. This way, F = G × K E is a (unimodular) vector bundle on X equipped with an invariant flat connection ∇ F,f and a Hermitian metric h F induced by h E . Moreover, (F, ∇ F,f , h F ) descends to a flat orbifold vector bundle on Z, which we still denote by the same notation. Let (Ω · (Z, F ), d Z,F ) be the de Rham complex associated with the orbifold vector bundle F over Z. It is equipped with an induced L 2 -metric. Let D Z,F,2 be the corresponding Hodge Laplacian.
For a closed manifold, the Ray-Singer real analytic torsion is defined in [32,33]. This notion extends naturally to the case of compact orbifolds. More precisely, set m = dim Z = dim X, for i = 0, · · · , m, let D Z,F,2 i be the restriction of D Z,F,2 on F -valued i-forms on Z. Let ϑ i (Z, F )(s) denote the the spectral zeta function of D Z,F,2 i . It is a meromorphic function in s ∈ C which is holomorphic at s = 0. Then the Ray-Singer real analytic torsion T (Z, F ) ∈ R is given by It is a (graded) spectral invariant of D Z,F,2 . Note that we fix the Hermitian metric h F here. In general, T (Z, F ) will depend on the choice of h F . If F is acyclic on Z, and if m is odd, then T (Z, F ) is independent of h F (cf. [38,Corollary 4.9]). We refer to [24], [38], etc for more details.
In this paper, we consider a special sequence of such flat vector bundles {F d } d∈N on Z, and we study the asymptotic behavior of T (Z, F d ) as d → +∞. When Z is a smooth manifold, i.e., Γ is torsion-free, such question is already studied by Müller [29] and by Müller-Pfaff [31,30]. When Z is a compact hyperbolic orbifold, such question is studied by Ksenia Fedosova in [14] using the method of harmonic analysis. Here, we consider this question for an arbitrary compact locally symmetric orbifold (of noncompact type).
The fundamental rank δ(G) (or δ(X)) of G (or X), by definition, is the difference of the complex rank of G and the complex rank of K. As we will see in Theorem 4.1.4, if δ(G) = 1, we always have (1.0.4) T (Z, F ) = 0.
If Γ is cocompact and torsion-free, this result is already known by [ [38,Corollary 5.4]. As a consequence of (1.0.4), we only need to deal with the case δ(G) = 1.
We now describe in brief the special sequence of flat vector bundles {F d } d∈N which is concerned here. After fixing a root data for U , let P ++ (U ) denote the set of (real) dominant weights of U . If λ ∈ P ++ (U ), let (E λ , ρ E λ ) be the irreducible unitary representation of U with the highest weight λ. We extend it to a representation of G. We require λ to be nondegenerate, i.e., as G-representations, (E λ , ρ E λ ) is not isomorphic to (E λ , ρ E λ •θ). We also take an arbitrary λ 0 ∈ P ++ (U ). If d ∈ N, let (E d , ρ E d , h E d ) be the unitary representation of U with highest weight dλ + λ 0 . By Weyl's dimension formula, dim E d is a polynomial in d.
Let {(F d , ∇ F d , h F d )} d∈N be the sequence of the corresponding (unimodular) flat vector bundles on X or on Z as described above. Note that we do not have a canonical choice of h E d (or h F d ) for each d ∈ N, but this makes no trouble here. Indeed, since λ is nondegenerate, for d large enough, we will have (1.0.5) H · (Z, F d ) = 0.
Furthermore, δ(G) = 1 implies that dim Z is odd. Then the different choices of h E d (or h F d ) give the same T (Z, F d ).
Let [Γ] denote the set of the conjugacy classes in Γ. If γ ∈ Γ is elliptic, we call [γ] ∈ [Γ] an elliptic class. Let E + [Γ] be the set of nontrivial elliptic classes (which is not the identity) in [Γ]. It is a finite subset. The first main result in this paper is the following theorem. Moreover, the degrees of P (d), P E [γ] (d) can be determined in terms of λ, λ 0 .
If Z is hyperbolic, i.e. G = Spin(1, 2n + 1), the above result was proved by Ksenia Fedosova in [14,Theorem 1.1]. If Γ is assumed to be torsion-free, i.e. E + [Γ] = ∅, then (1.0.6) was proved by Müller and Pfaff in [30,Theorem 1.1](also in [31, Theorem 1.1] for hyperbolic case). In particular, Müller and Pfaff gave more detail. They proved that there exists a polynomial P λ,λ0 (d) in d which depends only on λ, λ 0 and on X (i.e., independent of Γ) such that as d → +∞, They also showed that there exists a constant C λ,λ0 > 0 such that where R(d) is a polynomial in d of degree no greater than the degree of dim E d . Under our setting, Γ may contain torsion elements. Let S ⊂ Γ be the finite subgroup which acts on X trivially, then P (d) in Theorem 1.0.1 is just as follows (1.0.9) P (d) = Vol(Z) |S| P λ,λ0 (d), In fact, if ignoring that Z may be an orbifold, the term P (d) in (1.0.6) is the L 2 -torsion [26] associated with F d → Z. To compute P λ,λ0 (d), as in [30], we need to evaluate explicitly the identity orbital integrals of certain heat kernel.
Let D X,F d ,2 be the G-invariant Laplacian operator on X which is the lift of D Z,F d ,2 . For t > 0, let p X,F d t (x, x ′ ) denote the heat kernel of 1 2 D X,F d ,2 with respect to the Riemannian volume element on X. For t > 0, the identity orbital integral I X (E d , t) of p X,F d t is defined as follows, (1.0.10) where N Λ · (T * x X) is the number operator on Λ · (T * x X), and the right-hand side of (1.0.10) is independent of the choice of x ∈ X.
Let MI X (F d , s), s ∈ C denote the Mellin transform (cf. (7.3.53)) of I X (F d , t). It is a meromorphic function which is holomorphic at 0. Then set (1.0.11) PI X (F d ) = ∂ ∂s | s=0 MI X (F d , s).
Using essentially the Plancherel formula for I X (F d , t), Müller-Pfaff [30] managed to show that PI X (F d ) is a polynomial in d (for d large enough). In this paper, we use instead an explicit geometric formula of Bismut [3] for I X (F d , t) to compute PI X (F d ). In Subsection 7.4, we verify that our computational results coincide with the ones of Müller-Pfaff [30]. In (1.0.9), we just set (1.0.12) P λ,λ0 (d) = PI X (F d ).
Moreover, since X is of noncompact type with δ(X) = 1, by the de Rham decomposition of X, we have (as symmetric spaces) where X 2 is a symmetric space with δ(X 2 ) = 0, and X 1 is one of the following cases (1.0.14) X 1 = SL 3 (R)/SO (3) or SO 0 (p, q)/SO(p + q) with pq > 1 odd .
Correspondingly, there exist sequences of homogeneous flat vector bundles {F i,d } d∈N defined in the same way as F d but defined on the symmetric space X i , i = 1, 2, such that F d = F 1,d ⊠ F 2,d . In particular, {F 1,d } d∈N is associated with the dominant weights dλ 1 + λ 1,0 with certain nondegenerate λ 1 . By [ where [e(T X 2 , ∇ T X2 )] max2 is the top degree coefficient in the Euler form of X 2 , and rk(F 2,d ) is a polynomial in d. In [30, Proposition 6.7 & 6.8], they gave very explicit computations on PI X1 (F 1,d ) for the listed cases in (1.0.14). Therefore, we already have full power to deal with the term P (d) in Theorem 1.0.1. If G has a noncompact center with δ(G) = 1, but if it still admits a (linear) compact Lie group U as compact form, then the above consideration remains applicable. In particular, PI X (F d ) is well-defined as a polynomial in d. The concrete results are given in Corollary 7.3.7.
Our second main result of this paper is to give an explicit description on the term P E [γ] (d) appeared in Theorem 1.0.1. As an analogue of P (d), the term P E [γ] (d) will be an oscillating combination of the L 2 -torsions for the singular stratum of Z corresponding to [γ] ∈ E + [Γ]. Therefore, we will use a family of polynomials PI Xj (F j,d ) with δ(X j ) = 1 as bricks to construct an explicit formula for P E [γ] (d). We will also give a different method to evaluate it in Theorem 7.5.5.
For a clear statement of our results, we need introduce some notations and facts. Note that U contains K as a Lie subgroup. Let T be a maximal torus of K, and let T U be the corresponding maximal torus of U containing T .
Let X(k) denote the fixed point set of k acting on X. Then X(k) is a connected symmetric space with δ(X(k)) = 1. Let Z(k) 0 be the identity component of the centralizer Z(k) of k in G. Then X(k) = Z(k) 0 /K(k) 0 with K(k) 0 = Z(k) 0 ∩ K. Let U (k) denote the centralizer of k in U . Then U (k) 0 is naturally a compact form of Z(k) 0 . Then the triple (X(k), Z(k) 0 , U (k) 0 ) becomes a smaller version of (X, G, U ), except that Z(k) 0 may have noncompact center.
Let u be the Lie algebra of U , and let t U ⊂ u be the Lie algebra of T U . Let R(u, t U ) be the associated real root system with a system R + (u, t U ). Then P ++ (U ) ⊂ t * U is taken with respect to the above root data. Now we fix k ∈ T , and let u(k) denote the Lie algebra of U (k) 0 . Then T U is also a maximal torus of U (k) 0 . We get the following splitting of root systems where u ⊥ (k) is the orthogonal space of u(k) in u with respect to the Killing form. Let R + (u(k), t U ), R + (u ⊥ (k), t U ) be the induced positive root systems, and let ρ u , ρ u(k) denote the half of the sum of the roots in R + (u, t U ), R + (u(k), t U ) respectively. Let W (u C , t U,C ) be the Weyl group associated with the pair (u, t U ). Put , let ε(σ) denote its sign. If µ ∈ P ++ (U ), set where ξ α is the character of T U with (dominant) weight 2π √ −1α. Now we state our second main theorem. PE X,1 (F d ) = PI X (F d ).
(2) If γ = k ∈ T , for σ ∈ W 1 U (k), σλ is a nondegenerate dominant weight of U (k) 0 . Let E k σ,d denote the unitary representations of U (k) 0 (up to a finite central extension) with highest weight dσλ + σ(λ 0 + ρ u ) − ρ u(k) , d ∈ N, and let {F k σ,d } d∈N be the corresponding sequence of flat vector bundles on X(k). ϕ U k (σ, dλ + λ 0 )PI X(k) (F k σ,d ), where ϕ U k (σ, dλ + λ 0 ) is given by (1.0.18), and PI X(k) (F k d,σ ) is just the (real) polynomial in d defined via (1.0.11) with the sequence F k σ,d → X(k). Moreover, there exist constants C k σ > 0, σ ∈ W 1 U (k) such that (1.0.21) is an oscillating term of the form c 1 e 2π Theorem 1.0.1 now is just a consequence of (1.0. 22). Note that for [γ] ∈ E + [Γ], the (compact) orbifold Γ ∩ Z(γ)\X(γ) represents an orbifold stratum for the singular points of Z. An important observation on (1.0.22) is that the sequence {T (Z, F d )} d∈N encodes the volume information for Z and its singular strata.
In [31], the above observation is used to show that the volume of a compact odd-dimensional hyperbolic manifold Z is determined by the set of Reidemeister torsion invariants, since here T (Z, F d ) always coincides with the corresponding Reidemeister torsion (cf. [28]). In particular, the set of Reidemeister torsion invariants determines a compact hyperbolic 3-manifold up to a finitely many possibilities [29,Corollary 1.4]. Furthermore, in the context of manifolds, the result like (1.0.22) is also used to study the growth of the torsions in the cohomology. Such studies are carried out by Marshall and Müller [25] for arithmetic hyperbolic 3-manifolds with help of the results in [29]. Bergeron and Venkatesh [2] also studied the growth of torsions in (co)homology using this idea but under a totally different setting. However, in the context of orbifolds, the relation between the analytic torsions and topological torsions is sophisticated, so that we could not simply apply (1.0.22). Despite the above difficulty, we can still expect a potential application of (1.0.22) to study locally symmetric spaces, especially under arithmetic setting. Now we explain our approach to Theorem 1.0.2. Note that the result (2) in Theorem 1.0.2 follows from the representation theory, and (4) is an observation on the defining formula (1.0.18) of ϕ U k (σ, dλ + λ 0 ). Moreover, (1.0.21) is just a consequence of (1.0.8) proved in [30]. Therefore, we will concentrate on defining PE X,γ (F d ), where Tr s [·] denotes the supertrace with respect to the Z 2 -grading on Λ · (T * Z).
If γ ∈ G is semisimple, let E X,γ (F d , t) denote the orbital integral (cf. Subsection 3.3) of the Schwartz kernel of (N Λ · (T * X) − m 2 ) exp(−tD X,F d ,2 /2) associated with γ. Note that in E X,γ (F d , t), we should take the supertrace of the endomorphism on Λ · (T * X) ⊗ F as in (1.0.23). Moreover, E X,γ (F d , t) depends only on the conjugacy class of γ in G. Let ME X,γ (F d , s) denote the (formal) Mellin transform of E X,γ (F d , t), t > 0 with appropriate s ∈ C.
By applying the Selberg's trace formula to Z = Γ\X, we get Indeed, to handle the contribution of the nonelliptic [γ] ∈ [Γ], we use a spectral gap of D Z,F d ,2 due to the nondegeneracy of λ. By [6,Theorem 4.4] and [30,Section 7], there exist constants C > 0, c > 0 such that for d ∈ N, That is why (1.0.5) holds for d large enough. The Part 3 follows from a technical argument which makes good use of (1.0.26) and the fact that nonelliptic elements in Γ admit a uniform positive lower bound for their displacement distances on X. Now we consider [γ] ∈ E[Γ]. We will evaluate E X,γ (F d , t) by the geometric formula of Bismut [3, Theorem 6.1.1] for semisimple orbital integrals.
We can write E X,γ (F d , t) as an Gaussian-like integral with the integrand given as a product of an analytic function determined by the adjoint action of γ on Lie algebras and the character χ E d of the representation E d . By coordinating these two factors, especially using all sorts of character formulae for χ E d , we can integrate it out. We show that E X,γ (F d , t) is a finite sum of the terms as follows, where j ∈ N, c = 0, b are real constants, and Q(d) is a (nice) pseudopolynomial in d. It is crucial that c = 0. Indeed, we will see in Subsection 7.1 that the quantity c measures the difference between the representations which is a (nice) pseudopolynomial in d (for d large enough). We give the details on these computations in Subsections 7.3 and 7.5, where we apply the techniques inspired by the computations in Shen's approach to the Fried conjecture [37,Section 7].
In the same time, the formula (1.0.20) is a refined version of the above results on PE X,γ (F d ), since each PI X(k) (F k σ,d ) is already well understood. For proving it, we apply a geometric localization formula for E X,γ (F d , t) as follows.
Theorem 1.0.3. Assume that δ(G) = 1. We use the same notation as in Theorem 1.0.2. Let γ ∈ Γ be semisimple, up to a conjugation, we can write γ uniquely as a commuting product of its elliptic part k ∈ T and its hyperbolic part γ h ∈ G. Then there exists a constant c(γ) > 0 such that for t > 0, d ∈ N, The above theorem is restated as Theorem 6.0.1 in Section 6. When γ = k ∈ T , then γ h = 1 and c(γ) = 1. Then (1.0.28) reduces to After taking the Mellin transform on both sides of (1.0.29), we get exactly (1.0.20). Our approach to Theorem 1.0.3 is a more delicate application of Bismut's formula [3, Theorem 6.1.1]. As we said before, E X,γ (F d , t), E X(k),γ h (F k σ,d , t) are equal to integrals of some integrands involving χ E d , χ E k σ,d respectively. To relate the both sides of (1.0.28), we employ a generalized version of Kirillov character formula (cf. Theorem 5.4.4) which gives an explicit way of decomposing . This character formula was proved by Duflo, Heckman and Vergne in [12,Theorem (7)] under a general setting, and we will recall its special case for our need in Subsection 5.4. Then we expand the integral for E X,γ (F d , t) carefully, after replacing χ E d | U(k) 0 by the sum of χ E k σ,d , we rewrite this big integral as a sum of certain small integrals. Then we verify that each small integral is exactly some E X(k),γ h (F k σ,d , t) by Bismut's formula. This way, we prove (1.0.28). Theorem 1.0.3 can be interpreted as follows, the elliptic part k in γ could lead to a geometric localization onto its fixed point set X(k) when we evaluate the orbital integrals. Even thought we only prove it for a very restrictive situation, we still expect such phenomena in general due to a geometric formulation for the semisimple orbital integrals (cf. [3,Chapter 4], also Subsection 3.3).
Note that in [5,6], Bismut, Ma and Zhang studied the asymptotic analytic torsion forms in a more general context for compact manifold. In [6,Section 8], they also explained well how to use Bismut's formula for semisimple orbital integrals to compute the asymptotic analytic torsions.
This paper is organized as follows. In Section 2, we recall the definition of Ray-Singer analytic torsion for compact orbifolds. We also include a brief introduction to the orbifolds at beginning.
In Section 3, we introduce the explicit geometric formula of Bismut for semisimple orbital integrals and the Selberg's trace formula for compact locally symmetric orbifolds. They are the main tools to study the analytic torsions in this paper.
In Section 4, we give a vanishing theorem for T (Z, F ), so that we only need to focus on the case δ(G) = 1.
In Section 5, we study the Lie algebra of G provided δ(G) = 1. Furthermore, we introduce a generalized Kirillov formula any compact Lie groups.
In Section 6, we prove Theorem 1.0.3. In Section 7, given the sequence {F d } d∈N , we compute explicitly E X,γ (F d , t) in terms of root data for elliptic γ, in particular, we prove (1.0.27). Then we give formulae for PI X (F d ), PE X,γ (F d ).
In Section 8, we introduce the spectral gap (1.0.26) and we give a proof to Theorem 1.0.2.
In this paper, if V is a real vector spaces and if E is a complex vector space, we will use the symbol V ⊗ E to denote the complex vector space V ⊗ R E. If both V and E are complex vector spaces, then V ⊗ E is just the usual tensor over C. Acknowledgments. I would like to thank Prof. Jean-Michel Bismut, and Prof. Werner Müller for encouraging me to work on this subject, and for many useful discussions. I also thank Dr. Taiwang Deng for educating me about the cohomology of arithmetic groups, and Dr. Ksenia Fedosova for explanations of her results on the hyperbolic case.
This work is carried out during my stay in Max Planck Institute for Mathematics (MPIM) in Bonn. I also want to express my sincere gratitude to MPIM for providing so nice research environment.

Ray-Singer analytic torsion
In this section, we recall the definitions of the orbifold and the orbifold vector bundle. We also refer to [34,35]  π U is a continuous surjective U → U , which is invariant by G U -action; π U induces a homeomorphism between G U \ U and U .
If V ⊂ U is a connected open subset, an embedding of orbifold chart for the inclusion i : V → U is an orbifold chart ( V , π V , G V ) for V and an orbifold chart ( U , π U , G U ) for U together with a smooth embedding φ UV : V → U such that the following diagram commutes, two connected open subsets of Z with the charts ( U 1 , π U1 , G U1 ), ( U 2 , π U2 , G U2 ) respectively. We say that these two orbifold charts are compatible if for any point z ∈ U 1 ∩ U 2 , there exists an open connected neighborhood V ⊂ U 1 ∩U 2 of z with an orbifold chart ( V , π V , G V ) such that there exist two embeddings of orbifold charts φ U1V : An orbifold atlas (V, V) is called a refinement of (U, U ) if V is a refinement of U and if every orbifold chart in V has an embedding into some orbifold chart in U . Two orbifold atlas are said to be equivalent if they have a common refinement, and the equivalent class of an orbifold atlas is called an orbifold structure on Z.
An orbifold is a second countable Hausdorff space equipped with an orbifold structure. It is said to have dimension m if all the orbifold charts which defines the orbifold structure are of dimension m.
If Z, Y are two orbifolds, a smooth map f : Z → Y is a continuous map from Z to Y such that it lifts locally to an equivariant smooth map from an orbifold chart of Z to an orbifold chart of Y . In this way, we can define the notion of smooth functions and the smooth action of Lie groups.
By [38,Proposition 2.12], if Γ is discrete group acting smoothly and properly discontinuously on the left on an orbifold X, then Z = Γ\X has a canonical orbifold structure induced from X.
In the sequel, let Z be an orbifold with an orbifold structure given by (U, U). If z ∈ Z, there exists an open connected neighborhood U z of z with a compatible orbifold chart ( U z , G z , π z ) such that π −1 z (z) contains only one point x ∈ U z . Then G z does not depend on the choice of such open connected neighborhood (up to canonical isomorphisms compatible with the orbifold structure), then G z is called the local group at z. Put Then Z reg is naturally a smooth manifold. But Z sing is not necessarily an orbifold.
In [17,Section 2], the author provided two different methods to view Z sing as an immersed image of a disjoint union of orbifolds. We just recall that method which appears naturally in Kawasaki's local index theorems for orbifolds [17,18]. If z ∈ Z sing , let 1 = (h 0 z ), (h 1 z ), · · · , (h lz z ) be the conjugacy classes in G z . Put Let ( U z , G z , π z ) be the local orbifold chart for z ∈ Z sing such that π −1 z (z) contains only one point. For j = 1, · · · , l z , let U We say E to be an orbifold vector bundle of rank r on Z if there exists a smooth map of orbifolds π : E → Z such that for any U ∈ U and ( U , G U , π U ) ∈ U , there exists an orbifold chart ( U E , G E U , π E U ) of E such that U E is an vector bundle on U of rank r equipped an effective action of G E U and π E U ( U E ) = π −1 (U ). Moreover, there exists a surjective group morphism ψ U : Note that if E is proper, then the rank of E can be extended to a locally constant function ρ on ΣZ. The orbifold chart of Z i is given by the triples such as ( U z , h j z acts on the fibres U E z linearly, so that we can set ρ(z, (h j z )) = Tr U E z [h j z ]. One can verify this way, ρ is really a locally constant function on ΣZ. For i = 1, · · · , l, let ρ i be the value of ρ on the component Z i . We also put ρ 0 = r.
We call s : Z → E a smooth section of E over Z if it is a smooth map between orbifolds such that π • s = Id Z . We will use C ∞ (Z, E) to denote the vector space of smooth sections of E over Z.
Take an orbifold chart ( U , G U , π U ) ∈ U of Z, G U acts canonically on the tangent vector bundle T U of U . The open embeddings of orbifold charts of Z also lift to the open embeddings of their tangent vector bundles. This way, we get a proper orbifold vector bundle T Z on Z, and the projection π : T Z → Z is just given by the obvious projection T U → U . We call T Z the tangent vector bundle of Z. If we equipped T Z with Euclidean metric g T Z , we will call Z a Riemannian orbifold and call g T Z a Riemannian metric of Z.
Let Ω · (Z) denote the set of smooth differential forms of Z, which has a Z-graded structure by degrees. The de Rham operator d Z : Ω · (Z) → Ω ·+1 (Z) is given by the family of de Rham operator d U : Ω · ( U ) → Ω ·+1 ( U ). Then we can define the de Rham complex (Ω · (Z), d Z ) of Z and the associated de Rham cohomology H · (Z, R).
By [17,Section 1], there is a natural isomorphism between H · (Z, R) and the singular cohomology of the underlying topological space Z.
The Chern-Weil theory on the characteristic forms extends to orbifolds. We refer to [38,Section 3] for more details. Note that, as in [17,18], the characteristic forms are not only defined on Z but also defined on ΣZ. The part ΣZ has a nontrivial contribution in Kawasaki's local index theorems for orbifolds.
Finally, let us recall the integrals on Z. Assume that Z is compact. We may take a finite open covering {U i } i∈I of the precompact orbifold charts for Z. Since Z is Hausdorff, then there exists a partition of unity subordinate to this open cover. We can find a family of smooth functions By [38,Section 3.2], if α ∈ Ω m (Z, o(T Z)), then α is also integrable on Z reg , so that If (Z, g T Z ) is a Riemannian orbifold, we can define the integration on Z with respect to the Riemannian volume element. If we have a Hermitian orbifold vector bundle (F, h F ) → (Z, g T Z ), one can define the L 2 scalar product for the space of continuous sections of F as usual. Then we get the Hilbert space L 2 (Z, F ). If we also have a connection ∇ F , one also can define the Sobolev spaces of sections of F with respect to ∇ F and ∇ T Z . These constructions are parallel to the case of smooth manifolds. We refer [38, Section 3] for more details. Now we introduce the orbifold Euler characteristic number of (Z, g T Z ) [35]. The Euler form e(T Z, ∇ T Z ) ∈ Ω m (Z, o(T Z)) is given by the family of closed forms If Z is oriented, then we can view e(T Z, ∇ T Z ) as a differential form on Z.
If Z is compact, set By [35,Section 3], χ orb (Z) is a rational number, and it vanishes when Z is odd dimensional.

2.2.
Flat vector bundles and analytic torsions of orbifolds. If (F, ∇ F ) is an orbifold vector bundle over Z with a connection ∇ F , we call (F, ∇ F ) a flat vector bundle if the curvature R F = ∇ F,2 vanishes identically on Z. A detailed discussion for the flat vector bundles on Z is given in [38,Section 2.5].
Let (Z, g T Z ) be a compact Riemannian orbifold of dimension m. Let (F, ∇ F ) be a flat complex orbifold vector bundle of rank r on Z with Hermitian metric h F . Note that we do not assume that F is proper.
Let Ω · (Z, F ) be the set of smooth sections of Λ · (T * Z) ⊗ F on Z. Let d Z be the exterior differential acting on Ω · (Z, R).
Since ∇ F,2 = 0, then (Ω · (Z, F ), d Z,F ) is a complex, which is called the de Rham complex for the flat orbifold vector bundle (F, ∇ F ) on Z. Let H · (Z, F ) be its cohomology, which is called the de Rham cohomology of Z valued in F , as in the case of closed manifolds, H · (Z, F ) are always finite dimensional.
Let ·, · Λ · (T * Z)⊗F,z be the Hermitian metric on Λ · (T * z Z) ⊗ F z , z ∈ Z induced by g T Z z and h F z . Let dv be the Riemannian volume element on Z induced by g T Z . The L 2 -scalar product on Ω · (Z, F ) is given as follows, if s, s ′ ∈ Ω · (Z, F ), then By (2.1.8), it will be the same if we take the integrals on Z reg . Let d Z,F, * be the formal adjoint of d Z,F with respect to the L 2 -metric on Ω · (Z, F ), i.e., for s, s ′ ∈ Ω · (Z, F ), Then d Z,F, * is a first-order differential operator acting Ω · (Z, F ) on which decreases the degree by 1.
It is a first-order self-adjoint elliptic differential operator acting on Ω · (Z, F ).
The Hodge Laplacian is Here, [·, ·] denotes the supercommutator. Then D Z,F,2 is a second-order essentially self-adjoint positive elliptic operator, which preserves the degree. Let H 2 (Z, F ) be the Sobolev space of the bundle Λ · (T * Z) ⊗ F of order 2 with respect to ∇ T Z , ∇ F . Then the domain of the self-adjoint extension of D F,Z,2 is just H 2 (Z, F Ω · (Z, F ) = ker D Z,F,2 ⊕ Im(d Z,F | Ω · (Z,F ) ) ⊕ Im(d Z,F, * | Ω · (Z,F ) ).
Then we have the canonical identification of vector spaces, , whose elements are called harmonic forms of degree i. By Hodge theory, we have the canonical isomorphism of finite dimensional vector spaces for i = 0, 1, · · · , m, If F is proper, recall that the numbers ρ i , i = 0, · · · , l are defined in previous subsection as the extension of the rank of F . Then by [38,Theorem 4.3], we have The right-hand side of (2.2.11) contains the nontrivial contributions from ΣZ. The spectrum (with multiplicities) of D Z,F,2 i is of the form If s ∈ C, let ℜ(s) ∈ R denote its real part.
Definition 2.2.3. If i = 0, 1, · · · , m, the zeta function of D Z,F,2 i acting on Ω i (Z, F ) is defined as follows, if s ∈ C is such that ℜ(s) is big enough, By the standard heat equation method, one can see that ϑ i (F )(s) is well-defined for ℜ(s) > m 2 , and that ϑ i (F )(s) admits a meromorphic extension to s ∈ C which is holomorphic at s = 0. We also denote by ϑ i (s) this meromorphic extension. Then Definition 2.2.4. Let T (g T Z , ∇ F , h F ) ∈ R be given by The quantity T (g T Z , ∇ F , h F ) is called Ray-Singer analytic torsion associated with (F, ∇ F , h F ).
If m is even and if Z is orientable, then Let P = ⊕ i P i be the orthogonal projection from Ω · (Z, F ) to H · (Z, F ). Let H ⊥ denote the orthogonal subspace of H · (Z, F ) in Ω · (Z, F ), and let (D Z,F,2 ) −1 be the inverse of D Z,F,2 acting on H ⊥ . Let N Λ · (T * Z) be the number operator on Λ · (T * Z) which acts on Λ i (T * Z) by multiplication of i. Put By (2.2.14), (2.2.17), we get that if ℜ(s) is big enough, then For t > 0, as in [4, eq.(1.8.5)], put As t → +∞,

Orbital integrals and locally symmetric spaces
In this section, we recall some geometric properties of the symmetric space X, and we recall an explicit geometric formula of Bismut [3, Chapter 6] for semisimple orbital integrals. Then, given a cocompact discrete subgroup Γ ⊂ G, we describe the orbifold structure on Z = Γ\X, and we deduce in detail the Selberg's trace formula for Z.
In this section, G is taken as a connected linear real reductive Lie group, we do not require that it has a compact center. Then X is a symmetric space which may have de Rham components of both noncompact type and Euclidean type.
3.1. Real reductive Lie group. Let G be a connected linear real reductive Lie group with Lie algebra g, and let θ ∈ Aut(G) be a Cartan involution. Let K be the fixed point set of θ in G. Then K is a maximal compact subgroup of G, and let k be its Lie algebra. Let p ⊂ g be the eigenspace of θ associated with the eigenvalue −1. The Cartan decomposition of g is given by Put m = dim p, n = dim k. Let B be a G-and θ-invariant nondegenerate symmetric bilinear form on g, which is positive on p and negative on k. It induces a symmetric bilinear form B * on g * , which extends to a symmetric bilinear form on Λ · (g * ). The K-invariant bilinear form ·, · = −B(·, θ·) is a scalar product on g, which extends to a scalar product on Λ · (g * ). We will use | · | to denote the norm under this scalar product.
Let U g be the universal enveloping algebra of g. Let C g ∈ U g be the Casimir element associated with B, i.e., if {e i } i=1,··· ,m+n is a basis of g, and if {e * i } i=1,··· ,m+n is the dual basis of g with respect to B, then We can identify U g with the algebra of left-invariant differential operators over G, then C g is a second-order differential operator, which is Ad(G)-invariant.
Let z g ⊂ g be the center of g. Put Then They are orthogonal with respect to B. Let Z G be the center of G, let G ss be the analytic subgroup of G associated with g ss . Then G is the commutative product of Z G and G ss , in particular, For saving notation, if a ∈ p, we use notation ia ∈ √ −1p ⊂ u to denote the corresponding vector.
Then u is a (real) Lie algebra, which is called the compact form of g. Then Let G C be the complexification of G with Lie algebra g C . Then G is the analytic subgroup of G C with Lie algebra g. Let U ⊂ G C be the analytic subgroup associated with u. By [19,Proposition 5.3], if G has compact center, i.e. z(g) ∩ p = {0}, then U is a compact Lie group and a maximal compact subgroup of G C .
We call γ h = ge a g −1 , γ e = gkg −1 the hyperbolic, elliptic parts of γ. These two parts are uniquely determined by γ. If γ h = 1, we say γ to be elliptic, and if γ e = 1, we say γ to be hyperbolic.
Correspondingly, we have .
Then θ g is a Cartan involution of Z(γ) and B g is a nondegenerate symmetric bilinear form on z(γ). Let K(γ) be the fixed point set of θ g in Z(γ), then Moreover, K(γ) is a maximal compact subgroup of Z(γ), which meets every connected components of Z(γ). Let k(γ) ⊂ z(γ) be the Lie algebra of K(γ). Put Then the Cartan decomposition of z(γ) with respect to θ g is given by Moreover, B g is positive on p(γ), and negative on k(γ). The splitting in (3.1.14) is orthogonal with respect to B g .
Then X is a smooth manifold with the differential structure induced by G. By definition, X is diffeomorphism to p.
Let ω g ∈ Ω 1 (G, g) be the canonical left-invariant 1-form on G. Then by (3.1.1), Let p : G → X denote the obvious projection. Then p is a K-principal bundle over X. Then ω k is a connection form of this principal bundle. The associated curvature form If (E, ρ E , h E ) is a finite dimensional unitary or Euclidean representation of K, then F = G × K E is a Hermitian or Euclidean vector bundle over X with the unitary or Euclidean connection ∇ F induced by ω k . In particular, The bilinear form B restricting to p gives a Riemannian metric g T X , and ω k induces the associated Levi-Civita connection ∇ T X . Let d(·, ·) denote the Riemannian distance on X.
, so it induces an operator C g,X acting on C ∞ (X, F ). Let ∆ H,X be the Bochner Laplacian acting on C ∞ (X, F ) given by ∇ F , and let C k,E ∈ End(E) be the action of the Casimir C k on E via ρ E . The element C k,E induces an self-adjoint section of End(F ) over X. Then Let C k,p ∈ End(p), C k,k ∈ End(k) be the actions of Casimir C k acting on p, k via the adjoint actions. Moreover, we can view C k,p as a parallel section of End(T X). Let Ric X denote the Ricci curvature of (X, g T X ). By [3, Eq. (2.6.8)], If A ∈ End(E) commutes with K, then it can be viewed a parallel section of End(F ) over X. Let dx be the Riemannian volume element of (X, g T X ).
A be the Bochner-like Laplacian acting on C ∞ (X, F ) given by We will not distinguish the heat kernel p X t (x, x ′ ) and the function p X t (g) in the sequel.

3.3.
Bismut's formula for semisimple orbital integrals. The group G acts on X isometrically. If γ ∈ G, for x ∈ X, put It is called the displacement function associated with γ, which is a continuous convex function on X. Moreover, d 2 γ is a smooth convex function on X. By [13, Definition 2.19.21] and [3, Theorem 3.1.2], γ is semisimple if and only if d γ can reach its minimum m γ ≥ 0 in X. In particular, γ is elliptic if and only if γ has fixed points in X. If γ is semisimple, let X(γ) be the minimizing set of d γ , which is a geodesically convex submanifold of X.
If γ ∈ G is semisimple, then there exists g γ ∈ G such that Let Z(γ) 0 , K(γ) 0 be the connected components of the identity of Z(γ), K(γ) respectively. By [3, Theorem 3.3.1], Z(γ) 0 acts on X(γ) isometrically and transitively. Moreover, We equip the symmetric space Z(γ)/K(γ) with the Riemannian metric induced from B gγ | p(γ) , then the above identifications are isometric. Let dg be the left-invariant Haar measure on G induced by (g, ·, · ). Since G is unimodular, then dg is also right-invariant. Let dk be the Haar measure on K induced by −B k , then Let dy be the Riemannian volume element of X(γ), and let dz be the bi-invariant Haar measure on Z(γ) induced by B gγ . Let dk(γ) be the Haar measure on K(γ) such that Let Vol(K(γ)\K) be the volume of K(γ)\K with respect to dk, dk(γ). In particular, we have .
Let dv be the G-left invariant measure on Z(γ)\G such that is well-defined. As indicated by the notation, it only depends on the conjugacy class [γ] of γ in G.
In [3, Section 4.2], a geometric formula for Tr [γ] [exp(−tL X A )] is established. We explain it as follows. Recall that X(γ) is a totally geodesic submanifold of X on which Z(γ) acts isometrically and transitively. Let N X(γ)/X be the orthogonal normal bundle of X(γ) in X, and let N X(γ)/X denote its total space. Then N X(γ)/X ≃ X via the normal geodesics.
For x ∈ X(γ), let df be the Euclidean volume element on N X(γ)/X,x . Then there exists a positive function r It is clear that the right-hand side of (3.3.10) does not depend on the choice of x ∈ X(γ). Because of this geometric interpretation for Tr [γ] [exp(−tL X A )], we also call it a geometric orbital integral.
An explicit formula for Tr [γ] [exp(−tL X A )] is given in [3, Theorem 6.1.1], and an extension to the wave operators of L X A is given in [3, Section 6.3]. Now we describe in detail this formula of Bismut. We assume that (3.3.11) γ = e a k, a ∈ p, k ∈ K, Ad(k)a = a.
Let z ⊥ 0 , p ⊥ 0 , k ⊥ 0 be the orthogonal vector spaces to z 0 , p 0 , k 0 in g, p, k with respect to B. Then . Also the action ad(a) gives an isomorphism between , and it is an antisymmetric endomorphism with respect to the scalar product.
Recall that the function A is given by is a self-adjoint positive endomorphism. Put In (3.3.19), the square root is taken to be the positive square root.
Its square root is denoted by The value of (3.3.21) at Y k 0 = 0 is taken to be such that We recall the definition of the function J γ in [3, eq. (5.5.5)]. 3.4. Compact locally symmetric spaces. Let Γ be a cocompact discrete subgroup of G. Then Γ acts on X isometrically and properly discontinuously. Then Then Γ x is a finite subgroup of Γ. Put Then we always have r x > 0. Set Then S is a finite subgroup of Γ ∩ K, and a normal subgroup of Γ. Put Then Γ ′ acts on X effectively and we have Z = Γ ′ \X.
together with the action of Γ ′ on these charts give an orbifold structure for Z, so that Z = Γ\X is a compact orbifold with a Riemannian metric g T Z induced by g T X . By , and γ is elliptic if and only if γ ′ is elliptic.
Moreover, we have Note that the right-hand side of (3.4.9) is a disjoint union of compact orbifolds. If γ ′ ∈ Γ ′ , put .
follows. By definition in Subsection 2.1, we get the rest part of this proposition. This completes the proof.
Note that Γ\G is a compact smooth homogeneous space equipped with a right action of K. Moreover, the action of K is almost free, i.e. for eachḡ ∈ Γ\G, the stabilizer Kḡ is finite. Then the quotient space (Γ\G)/K also have a natural orbifold structure, which is equivalent to Z. Indeed, givenḡ ∈ Γ\G, we get a unique Cartan decomposition of g = e f k with f ∈ p, k ∈ K, then C(k −1 )S ⊂ Kḡ. One can verify that S represents the isotropy group of the principal orbit type for the right action of K on Γ\G. If B p (0, ε) is a small enough open ball in p centered at 0, thenḡB p (0, ε) is an equivariant slice atḡ with respect to the action of K. Then the orbifold charts of (Γ\G)/K are given by the effective right action of C(k −1 )S\Kḡ onḡB p (0, ǫ). If x = pg ∈ X, then the adjoint action of Kḡ on B p (0, ǫ) is equivalent to the action of Γ x on the corresponding open ball in X centered x. This implies exactly that Z = (Γ\G)/K. Set Recall that Z G is the center of G. Using the fact that Γ is discrete, we get Moreover, Let dḡ be the volume element on Γ\G induced by dg. By (3.3.5), we get Example 3.4.2 (An non-natural example of the case S = ∆). Let G, K, θ, Γ be as before. Take K ′ a connected compact linear group which is center free, and let Then the corresponding S = S × Σ. But the group ∆ = ∆. This way, we can construct many examples such that S = ∆.
We only need to prove that in this case, S ⊂ Z G . Since G is connected, it is enough to prove that if s ∈ S, the adjoint action of s on g is trivial.
The action of θ preserves the splitting in (3.1.4). Let g ss = p ss ⊕k ss be the Cartan decomposition of g ss with respect to θ, where k ss ⊂ k, p ss ⊂ p. Moreover, since G ss is noncompact, then p ss is nonzero.
Since g ss is simple, then we have (3.4.16) [k ss , p ss ] = p ss , [p ss , p ss ] = k ss .
If s ∈ S, then Ad(s) acts trivially on p ss , thus it acts trivially on g ss and g. This completes the proof of our lemma.
Corollary 3.4.4. If g ss has no compact simple factor, then We note that in many interesting cases, we can reduce to the case of S = {1}. For instance, given a Riemannian symmetric space (X, g T X ) of noncompact type, let G = Isom(X) 0 be the connected component of identity of the Lie group of isometries of X. By [13, Proposition 2.1.1], G is a semisimple Lie group with trivial center. We refer to [13,Chapter 2] and [3, Chapter 3] for more details. This way, any subgroup of G acts on X effectively. In particular, if Γ is a cocompact discrete subgroup of G, then Z = Γ\X is a compact good orbifold with the orbifold fundamental group Γ. By (3.4.9), we have In general, by [16, Ch.V §4, Theorem 4.1], G = Isom(X = G/K) 0 if and only if K acts on p effectively. Another particular case is that S = ∆. Then after replacing G, K by their quotients G/S, K/S, we also go back to the case of S = {1}.
If ρ : Γ ′ → GL(C k ) is a representation of Γ ′ , which can be viewed as a representation of Γ via the projection Γ → Γ ′ = Γ/S, then F = X Γ ′ × C k is a proper flat orbifold vector bundle on Z with the flat connection ∇ F,f induced from the exterior differential d X on C k -valued functions. By [38,Theorem 2.31], all the proper orbifold vector bundle on Z of rank k comes from this way. Now let ρ : Γ → GL(C k ) be a representation of Γ, we do not assume that it comes from a representation of Γ ′ . We still have a flat orbifold vector bundle (F, We have the following results. In particular, if ρ| S : S → GL(C k ) does not have the isotypic component of the trivial representation of S on C, then Let (E, ρ E ) be a finite dimensional complex representation of G. When restricting to Γ, K, we get the corresponding representations of Γ, K respectively, which are still denoted by ρ E .
Set F = G × K E. Then F is a homogeneous vector bundle on X discussed in Subsection 3.2. Moreover, G acts on F . Then it descends to an orbifold vector bundle on Z.
The map (g, v) ∈ G× K E → (pg, ρ E (g)v) ∈ X ×E gives a canonical trivialization of F over X. This identification gives a flat connection ∇ F,f for F . Recall that the connection ∇ F is induced from ω k . Then This flat connection is G-invariant on X, then it descends to a flat connection on the orbifold vector bundle F on Z. Moreover, the above trivialization of F → X implies that the flat orbifold vector bundle (F, ∇ F,f ) is exactly the one given by For t > 0, let p Z t (z, z ′ ), z, z ′ ∈ Z be the heat kernel of L Z A over Z with respect to dz ′ . If z, z ′ are identified with their lifts in X, then If F is a proper orbifold vector bundle on Z, i.e., S acts trivially on E, then Γ ′ acts on F → X, and (3.5.1) can be rewritten as For the case where F is not a proper orbifold vector bundle, it will be more complicated. If g ∈ G, then g −1 Sg is also a subgroup of K. Put E pr g be the maximal subspace of E on which g −1 Sg acts trivially via ρ E . Then if k ∈ K, we have Then all the pairs (g, E pr g ), g ∈ G defines a subbundle of F on Z, which is just the corresponding proper orbifold bundle F pr of F . As in Proposition 3.4.5, we have If we write (E, ρ E | S ) as a direct sum of the trivial S-representation E pr 1 and the nontrivial part Combining (3.5.1), (3.5.4) and (3.5.5), we see that (3.5.2) still holds as kernels of integral operators acting on C ∞ (Z, F ).
As explained before, many interesting cases have the property S = 1 or can be reduced to S = 1. In these cases, the trace formula (3.5.17) shows clearly the different contributions from Z and from each components of ΣZ. In the sequel, we give a precise description of the contribution of ΣZ when S is not such trivial.
The orbifold resolution ΣZ of Z sing is described by (3.4.9). Recall that |S ′ (γ ′ )| is the multiplicity of the connected component Z Γ ′ (γ ′ )\X(γ ′ ) in ΣZ. Let π Γ : Γ → Γ ′ denote the obvious projection. If γ ′ ∈ Γ ′ , put Then (3.5.21) Proof. It is clear that (3.5.21) is just a special case of (3.5.22). But for a better understanding, we also include an easy proof to this special case, then we will prove the general case. If s ∈ S, then Ad(s) acts trivially on p. Then we have the identification Therefore, we get the following identification together with their volume elements, Then the identity (3.5.20) follows from (3.2.10), (3.3.9) and the fact that S acts trivially both on X. Now we prove (3.5.21). By (3.5.16) Then by (3.5.24), Then Now we prove the second part. Let γ 0 ∈ Γ be such that π Γ (γ 0 ) ∈ [γ ′ ] ′ . Then Also note that the quantity Tr [γ0·] [exp(−tL X A )] S is independent of the choice of such γ 0 , which is determined uniquely by [γ ′ ] ′ . By (3.5.10), (3.5.13) and (3.5.16), we get Note that X can be identified with the total space of the orthogonal normal bundle of X(γ ′ ) = X(γ 0 ) = X(γ 0 s). Moreover Z Γ ′ (γ ′ ) acts on X(γ ′ ) isometrically, and Z(γ 0 s) acts on X(γ ′ ) transitively. Then we get

Analytic torsions for compact locally symmetric spaces
In this section, we explain how to make use of Bismut's formula (3.3.25) and Selberg's trace formula (3.5.17) to study the analytic torsions of Z. We continue using the same settings as in Section 3.
We show that by a vanishing result on the analytic torsion, only the case δ(G) = 1 remains interesting. Then if G has noncompact center, we can get very explicit information for evaluating T (Z, F ) due to the Euclidean component in X. If G has compact center, we need more tools, which will be carried out in Sections 5 & 6.

4.1.
A vanishing result on the analytic torsions. Recall that G is a connected linear real reductive Lie group. Recall that z g is the center of g. Set Then Let T be a maximal torus of K with Lie algebra t, put It is clear that Put h = b ⊕ t, then h is a Cartan subalgebra of g. Let H be analytic subgroup of G associated with h, then it is also a Cartan subgroup of G. Moreover, dim t is just the complex rank of K, and dim h is the complex rank of G.
Definition 4.1.1. Using the above notations, the deficiency of G, or the fundamental rank of G is defined as The number m − δ(G) is even.
The following result is proved in [37,Proposition 3.3].
The two sides of (4.1.6) are equal if and only if γ can be conjugated into H.
Recall that u = √ −1p ⊕ k is the compact form of G, and that U ⊂ G C is the analytic subgroup with Lie algebra u. Let U u, U g C be the enveloping algebras of u, g C respectively. Then U g C can be identified with the left-invariant holomorphic differential operators on G C . Let C u ∈ U u be the Casimir operator of u associated with B, then In the sequel, we always assume that U is compact, this is the case when G has compact center. We now fix a unitary representation (E, ρ E , h E ) of U , and we extend it to a representation of G, whose restriction to K is still unitary.
Put F = G × K E with the Hermitian metric h F induced by h E . Let ∇ F be the Hermitian connection induced by the connection form ω k . If G has compact center, then F is a unimodular.
Furthermore, as explained in the last part of Subsection 3.4, F is equipped with a canonical flat connection ∇ F,f as follows, Let (Ω · c (X, F ), d X,F ) be the (compactly supported) de Rham complex twisted by F . Let d X,F, * be the adjoint operator of d X,F with respect to the L 2 metric on Ω · c (X, F ). The Dirac operator D X,F of this de Rham complex is given by (4.1.9) D X,F = d X,F + d X,F, * .
The Clifford algebras c(T X), c(T X) act on Λ · (T * X). We still use e 1 , · · · , e m to denote an orthonormal basis of p or T X, and let e 1 , · · · , e m be the corresponding dual basis of p * or T * X.
Let ∇ Λ · (T * X)⊗F,u be the unitary connection on Λ · (T * X) ⊗ F induced by ∇ T X and ∇ F . Then the standard Dirac operator is given by c(e j )ρ E (e j ).
In the same time, C g descends to an elliptic differential operator C g,X acting on C ∞ (X, Λ · (T * X) ⊗ F ). Let κ g ∈ Λ 3 (g * ) be such that if a, b, c ∈ g, Then κ g is a G-invariant closed 3-form on G. The bilinear form B induces a corresponding bilinear form B * on Λ · (g * ). Let C k be the Casimir operator associated with (k, B| k ), and let C k,k , C k,p be the the actions of C k on k, p via adjoint actions. By [3, Eq.(2.6.11)], we have Set (4.1.14) By [6,Proposition 8.4], we have Let γ ∈ G be a semisimple element. In the sequel, we may assume that (4.1.16) γ = e a k, a ∈ p, k ∈ K, Ad(k)a = a.
We also use the same notation as in Subsection 3.3.
Recall that p = dim p(γ), q = dim k(γ  Now we take a cocompact discrete subgroup Γ ⊂ G. Then Z = Γ\X is a compact locally symmetric orbifold. We use the same notation as in Subsections 3.4 & 3.5. Then we get a flat orbifold vector bundle (F, ∇ F,f , h F ) on Z. Furthermore, D X,F descends to the corresponding Hodge-de Rham operator D Z,F acting on Ω · (Z, F ). Let T (Z, F ) denote the associated analytic torsion as in Definition 2.2.4, i.e., As explained in Subsection 2.2, for computing T (Z, F ), it is enough to evaluate Then we apply the Selberg's trace formula in Theorem 3.5.2. We get   Therefore, the only nontrivial case is that δ(G) = 1, so that m is odd. If γ ∈ G is of the form (4.1.16). Let t(γ) ⊂ k(γ) be a Cartan subalgebra. Put In particular, a ∈ b(γ). Then h(γ) = b(γ) ⊕ t(γ) is a Cartan subalgebra of z(γ).
Recall that H is a maximal compact Cartan subgroup of G. The following result is just an analogue of [37,Theorem 4.12] and [3, Theorem 7.9.1].
Proposition 4.1.5. If δ(G) = 1, if γ is semisimple and can not be conjugated into H by an element in G, then Proof. Let t be a Cartan subalgebra of k containing t(γ).
This implies (4.1.24). The proof is completed.

Set
Then g ′ is an ideal of g. Let G ′ be the analytic subgroup of G associated with g ′ , which has a compact center. The group K is still a maximal subgroup of G ′ . Let U ′ ⊂ U be the compact form of G ′ with Lie algebra u ′ , then Now we assume that δ(G) = 1 and that G has noncompact center, so that b = z p has dimension 1. Then δ(G ′ ) = 0. Under the hypothesis that U is compact, then up to a finite cover, we may write We take a 1 ∈ b with |a 1 | = 1. If (E, ρ E ) is an irreducible unitary representation of U , then ρ E (a 1 ) acts on E by a real scalar operator. Let α E ∈ R be such that Put X ′ = G ′ /K. Then X ′ is an even-dimensional symmetric space (of noncompact type). We identify z p with a real line R, then In this case, the evaluation for analytic torsions can be made more explicit. If γ ∈ G ′ , let X ′ (γ) denote the minimizing set of d γ (·) in X ′ , so that Let [·] max denote the coefficient of a differential form on X ′ of the (oriented) Riemannian volume form. Similarly, for k ∈ T , let [·] max(k) denote the coefficient of the Riemannian volume form in a differential form on X ′ (k).
The following results are the analogue of [37, Proposition 4.14].
Proposition 4.1.6. Assume that G has noncompact center with δ(G) = 1 and that (E, ρ E ) is irreducible. Then (4.1.32) If γ = e a k is such that a ∈ b, k ∈ T , then If γ can not be conjugated into H, then Proof. Let C u ′ denote the Casimir operator of u ′ associated with B| u ′ . Then we have C u,E = −α 2 E + C u ′ ,E Then by (4.1.36) and [6,Theorem 8.5], an modification of the proof to [37,Proposition 4.14] proves the identities in our proposition. Note that (4.1.34) is just a special case of (4.1.24).
If we assembly the results in Proposition 4.1.6, it is enough to study the corresponding analytic torsion in most cases. We will get back to this point in Corollary 7.3.7 for asymptotic analytic torsions.

4.2.
Symmetric spaces of noncompact type with rank 1. In this subsection, we focus on the case where δ(G) = 1 and G has compact center (i.e. z p = 0), so that X is a symmetric space of noncompact type [37,Proposition 6.18].
Let U 1 , U 2 be (connected linear) compact forms of G 1 , G 2 . Then we may take U = U 1 × U 2 . Let (E, ρ E ) be an irreducible unitary representation of U , then where (E j , ρ Ej ) is an irreducible unitary representation of U j , j = 1, 2. Let F , F 1 , F 2 be the homogeneous flat vector bundles on X, X 1 , X 2 associated with these representations. Then we have Furthermore, γ is semisimple (resp. elliptic) if and only if both γ 1 , γ 2 are semisimple (resp. elliptic). Set m i = dim X i , then m 2 is even.
Proof. Note that the orbital integrals here are multiplicative with respect to the products of the underlying Lie groups. We write Note that, since δ(G 1 ) = 1, then by [  2). But it is far from enough to get explicit evaluations. In Sections 5 & 6, we will carry out more tools, which allows us work out a proof to Theorem 1.0.2.

Cartan subalgebra and root system of G when δ(G) = 1
We use the same notation as in previous sections. In Subsections 5.1 -5.3, we assume that G has compact center with δ(G) = 1. But, as we will see in Remark 5.3.3, the constructions and results in these subsections are still true (most of them are trivial) if U is compact and if G has noncompact center with δ(G) = 1.
Recall that T is a maximal torus of K with Lie algebra t ⊂ k, and that b ⊂ p is defined in (4.1.3). Since δ(G) = 1, then b is 1-dimensional. We now fix a vector a 1 ∈ b, |a 1 | = 1. Recall that h = b ⊕ t is a Cartan subalgebra of g. Let h g C be the Hermitian product on g C induced by the scalar product −B(·, θ·) on g.

5.1.
Reductive Lie algebra with fundamental rank 1. Since G has compact center, then b ⊂ z g .
Let Z(b) be the centralizer of b in G, and let Z(b) 0 be its identity component with Lie algebra Then m is a Lie subalgebra of z(b), which is invariant by θ. Put be the orthogonal subspaces of z(b), p(b), k(b) in g, p, k respectively with respect to B. Then Moreover, Let M ⊂ Z(b) 0 be the analytic subgroup associated with m. If we identify b with R, then  .
is also an isomorphism of K M -modules. Since θ fixes K M , n ≃n as K M -modules via θ.
By [37, Proposition 6.3], we have Then the bilinear form B induces an isomorphism of n * andn as K M -modules. Therefore, as K M -modules, n is isomorphic to n * . As a consequence of (5.1.10), we get Then (g, z(b)) is a symmetric pair. If k ∈ K M , let M (k) be the centralizer of k in M , and let m(k) be its Lie algebra. Let M (k) 0 be the identity component of M (k). The Cartan involution θ acts on M (k). The associated Cartan decomposition is Recall that Z(k) be the centralizer of k in G and that Z(k) 0 is the identity component of Z(k) with Lie algebra z(k) ⊂ g. Then Note that Z(k) 0 is still a reductive Lie group equipped with the Cartan involution induced by the action of θ. By the assumption that δ(G) = 1, we have In particular, We also have the following identities, Since δ(m(k)) = 0, dim n(k) is even. We set Since θ fixes K M (k), n(k) ≃n(k) as K M (k)-modules via θ.

A compact Hermitian symmetric space
Let u(b) ⊂ u, u m ⊂ u be the compact forms of z(b), m. Then Since M has compact center, let U M be the analytic subgroup of U associated with u m . Then U M is the compact form of M . Let U (b) ⊂ U , A 0 ⊂ U be the connected subgroups of U associated with Lie algebras u(b), √ −1b. Then A 0 is in the center of U (b). By [37, Proposition 6.6], A 0 is closed in U and is diffeomorphic to a circle S 1 . Moreover, we have The bilinear form −B induces an Ad(U )-invariant metric on u. Let u ⊥ (b) ⊂ u be the orthogonal subspace of u(b). Then By (5.1.12), we get Then (u, u(b)) is a symmetric pair. Put a 0 = a 1 /β(a 1 ) ∈ b. Set By (5.1.9), J is an U (b)-invariant complex structure on u ⊥ (b) which preserves B| u ⊥ (b) . The spaces n C = n ⊗ R C,n C =n ⊗ R C are exactly the eigenspaces of J associated with eigenvalues √ −1, − √ −1. The following proposition is just the summary of the results in [37, Section 6B].
Then Y b is a compact symmetric space, and J induces an integrable complex structure on Y b such that Remark 5.2.2. By [37,Proposition 6.20], if G has compact center, then as symmetric spaces, the Kähler manifold Y b is isomorphic either to SU(3)/U(2) or to SO(p + q)/SO(p + q − 2) × SO(2) with pq > 1 odd. This way, the computations on Y b can be made explicitly.
Let ω u be the canonical left-invariant 1-form on U with values in u. Let ω u(b) and ω u ⊥ (b) be the u(b) and u ⊥ (b) components of ω u , so that

Moreover, ω u(b) defines a connection form on the principal
Let Ω u(b) be the curvature form, then Note that the real tangent bundle of Y b is given Recall that the first splitting in (5.2.1) is orthogonal with respect to −B. Let Ω um be the u m -component of Ω u(b) . By [37, eq.(6-48)], Moreover, by [37, Proposition 6.9], we have Now we fix k ∈ K M . Let U (k) be the centralizer of k in U , and let U (k) 0 be its identity component. If k is not of finite order, then U (k) = U (k) 0 . Let u(k) be the Lie algebra of U (k) 0 . Then u(k) is the compact form of z(k), and U (k) 0 is the compact form of Z(k) 0 .
We will use the same notation as in Subsection 5.1. Then the compact form of m(k) is given by Let u b (k) be the compact form of z b (k). Then Let U b (k) be the analytic subgroup associated with u b (k). Then As in Proposition 5.
. Then the real tangent bundle of Y b (k) is given by Let Ω u b (k) be the curvature form as in (5.2.9) for the pair (U (k) 0 , U b (k)), which can be viewed as an element in Λ 2 (u ⊥ b (k) * ) ⊗ u b (k). Using the splitting (5.2.14), let Ω um(k) be the u m (k)-component of Ω u b (k) . Then as in (5.2.11) and (5.2.12), we have (5.2.20)

Positive root system and character formula.
Recall that t is Cartan subalgebra of k, of k m , and of m.
Let H ⊂ G be the analytic subgroup associated with h. Then The group H is a maximally compact Cartan subgroup of G. Put Then t U is a Cartan subalgebra of u. Let T U ⊂ U be the corresponding maximal torus. Then A 0 is a circle in T U . Then t is a Cartan subalgebra of u m , and the corresponding maximal torus is T . We have t U,C = h C . Let R(u C , t U,C ) be the associated root system. Then Similarly, we have .
. Fix a positive root system R + (m C , t C ), we get a positive root system R + (g C , h C ) consisting of element α such that α(a 1 ) > 0 and the elements in R + (m C , t C ).
Put W g = W (g C , h C ) = W (u C , t U,C ) the associated Weyl group. If ω ∈ W g , let l(ω) denote the length of ω with respect to R + (g C , h C ). Set Let W (U, T U ) be the analytic Weyl group, then W g = W (U, T U ). Put Let R(u, t U ) be the real root system for the pair (U, T U ) [9, Chapter V]. Then if α 0 ∈ R(u, t U ), after tensoring with C, we view it as an element in t * U,C , then 2πiα 0 ∈ R(u C , t U,C ). If α ∈ R(u C , t U,C ), then α 0 = α 2πi | tU ∈ R(u, t U ). Similarly, let R(u(b), t U ), R(u m , t) denote the real root systems for the pairs (u(b), t U ), (u m , t). When we embed t * into t * U by the splitting in (5.3.2), then The relations of them to the complex root systems are the same as above.
Then ρ u | t = ρ um . Let P ++ (U ) ⊂ t * U be the set of dominant weights of (U, T U ) with respect to R + (u, t U ). If λ ∈ P ++ (U ), let (E λ , ρ E λ ) be the irreducible unitary representation of U with the highest weight λ, which by the unitary trick extends to an irreducible representation of G.
Recall that U (b) acts on n C . Let H · (n C , E λ ) be the Lie algebra cohomology of n C with coefficients in E λ (cf. [21]) By [39, Theorem 2.5.1.3], for i = 0, · · · , 2l, we have the identification of U (b)-modules, By (5.3.9) and the Poincaré duality, we get the following identifications as U (b)modules, Note that if we apply the unitary trick, the above identification also holds as Z(b) 0modules.
Note that is a dominant weight of (U M , T ) with respect to R + (u m , t). Moreover, the restriction of the U (b)-representation V λ,ω to the subgroup U M is irreducible, which has the highest weight η ω (λ).

Proof.
Since ω(λ + ρ u ) − ρ u is analytically integrable, then η ω (λ) is also analytically integrable as a weight associated with (U M , T ). By (5.3.7) and the corresponding identification of positive root systems, we know that η ω (λ) is dominant with respect to R + (u m , t).
Recall that A 0 ≃ S 1 is defined in Subsection 5.2. The group U (b) has a finite extension A 0 × U M , then we view V λ,ω as an irreducible unitary representation of A 0 × U M , whose restriction to U M is clearly irreducible with highest weight η ω (λ). This completes the proof of our proposition.
Remark 5.3.3. In general, U is just the analytic subgroup of G C with Lie algebra u. If U is compact but G has noncompact center, i.e., z p = b, then n =n = 0, so that l = 0. Recall that in this case, G ′ , U ′ are defined in Subsection 4.1. Then we have The compact symmetric space Y b now reduces to one point. Moreover, in (5.3.6), the set W u = {1}, so that V λ,ω becomes just E λ itself. The identities (5.3.9), (5.3.10) are trivially true, so is Proposition 5.3.2.

5.4.
Kirillov character formula for compact Lie groups. In this subsection, we recall the Kirillov character formula for compact Lie groups. We only use the group U M as an explanatory example. We fix the maximal torus T and the positive (real) root system R + (u m , t).
Let λ ∈ t * be a dominant (analytically integrable) weight of U M . Let (V λ , ρ V λ ) be the irreducible unitary representation of U M with the highest weight λ.
Since λ + ρ um is regular, then we have the following identifications of U Mmanifolds, Then a L is a tangent vector field on O λ+ρu m . Such vector fields span the whole tangent space at each point.
Then ω L is a U M -invariant symplectic form on O λ+ρu m . Put r + = 1 2 dim u m /t. In fact, if we can define an almost complex structure on T O λ+ρu m such that the holomorphic tangent bundle is given by the positive root system R + (u m , t). Then (O λ+ρu m , ω L ) become a closed Kähler manifold, and r + is just its complex dimension.
The Liouville measure on O λ+ρu m is given as follows, It is invariant by the left action of U M . By the Kirillov formula, if y ∈ u m , we have If k ∈ T , put Z = U M (k) 0 , then T ⊂ Z. Then T is also a maximal torus of Z. Let z ⊂ u m be the Lie algebra of Z.
In the sequel, we will give a generalized version of (5.4.6) for describing the function Tr V λ [ρ V λ (ke y )] with y ∈ z.
Let q be the orthogonal space of z in u m with respect to B, so that Let R(z, t) be the real root system associated with the pair (z, t). Since the adjoint action of T preserves the splitting in (5.4.7). Then R(u m , t) splits into two disjoint parts where R(q, t) is just the set of real roots for the adjoint action of t on q C . The positive root system R + (u m , t) induces a positive root system R + (z, t). Set Then we have the disjoint union as follows, Put (5.4.11) ρ z = 1 2 Then (5.4.12) Let C ⊂ t * denote the Weyl chamber corresponding to R + (u m , t), and let C 0 ⊂ t * denote the Weyl chamber corresponding to R + (z, t). Then C ⊂ C 0 .
Let W (u m,C , t C ), W (z C , t C ) be the Weyl groups associated with the pair (u m , t), (z, t) respectively. Then W (z C , t C ) is canonically a subgroup of W (u m,C , t C ). Put Note that the set W 1 (k) is similar to the set W u defined in (5.3.6).
Lemma 5.4.1. The inclusion W 1 (k) ֒→ W (u m,C , t C ) induces a bijection between W 1 (k) and the quotient W (z C , t C )\W (u m,C , t C ).
Proof. This lemma follows from that W (z C , t C ) acts simply transitively on the Weyl chambers associated with (z, t).
Let λ ∈ t * be the dominant weight of U M as before. Then λ + ρ um ∈ C. Set The following lemma can be found in [12, Lemma (7) where the union is disjoint.
Note that if y ∈ z, the following analytic function has a square root which is analytic in y ∈ z and equals to 1 at y = 0. We denote this square root by The following theorem is a special case of a generalized Kirillov formula obtained by Duflo, Heckman and Vergne [12,Theorem (7)]. We will also include a proof for the sake of completeness.
Theorem 5.4.4 (Generalized Kirillov formula). For y ∈ z, we have the following identity of analytic functions, Proof. Let t ′ denote the set of regular element in t associated with the root R(u m , t), which is an open dense subset of t. Since both sides of (5.4.21) are invariant by adjoint action of Z, then we only need to prove (5.4.21) for y ∈ t ′ . We firstly compute the left-hand side of (5.4.21). For y ∈ t ′ , then We have Let y 0 ∈ t be such that k = exp(y 0 ). Then By the Weyl character formula for (U M , T ), we get = ω∈W (u m,C ,t C ) ε(ω)e 2πi ω(λ+ρu m ),y+y0 Π α 0 ∈R + (um,t) e πi α 0 ,y+y0 − e −πi α 0 ,y+y0 . Note that since 2ρ z is analytically integrable, then (5.4.27) ξ 2ρz (k) = 1.
We now prove (5.4.30). Let c be the center of u m , and put By the Weyl's theorem [19,Theorem 4.26], the universal covering group of U M,ss is compact, which we denote by U M,ss . Put Then U M is canonically a finite covering of U M . Let T be the maximal torus of U M associated with the Cartan subalgebra t. Let k = exp(y 0 ) ∈ T be a lift of k ∈ T . Let Z be the analytic subgroup of U M associated with z.
Note that Ad( k)| um = Ad(k)| um , then Z is a finite cover of Z. Let N Z ( T ) be the normalizer of T in Z, then The weight ρ um is analytically integrable with respect to T . Then by (5.4.35), if ω ∈ W (z C , t C ), (5.4.36) ξ ρu m ( k) = ξ ωρu m ( k).
Combining together (5.4.37) -(5.4.39), a direct computation shows that the right-hand side of (5.4.21) is given exactly by (5.4.29). This completes the proof of our theorem.
Remark 5.4.5. Note that for σ ∈ W 1 (k), the regular positive weight σ(λ + ρ um ) is analytically integrable with respect to ( Z, T ). If ρ z is also analytically integrable with respect to T , then σ(λ + ρ um ) − ρ z is a dominant weight for ( Z, T ) with respect to R + (z, t). In this case, let E k σ = E σ(λ+ρu m )−ρz be the irreducible unitary representation of Z with highest weight σ(λ + ρ um ) − ρ z . Then by (5.4.6), (5.4.21), we get that for y ∈ z, (5.4.40) Let Vol L (O k σ(λ+ρu m ) ) denote its symplectic volume with respect to the Liouville measure. Then Note that the first equality of (5.4.41) still holds even ρ z is not analytically integrable with respect to T .

A geometric localization formula for orbital integrals
Recall that G C is the complexification of G with Lie algebra g C , and that G, U are the analytic subgroups of G C with Lie algebra g, u respectively. In this section, we always assume that U is compact, we do not require that G has compact center. We need not to assume δ(G) = 1 either.
Under the settings in Subsection 4.1, for t > 0 and semisimple γ ∈ G, we set The indice X, F in this notation indicate precisely the symmetric space and the flat vector bundle which are concerned for defining the orbital integrals. If γ ∈ G is semisimple, then there exists a unique elliptic element γ e and a unique hyperbolic element γ h in G, such that γ = γ e γ h = γ h γ e . Here, we will show that E X,γ (F, t) becomes a sum of the orbital integrals associated with γ h , but defined for the centralizer of γ e instead of G. This suggests that the elliptic part of γ should lead to a localization for the geometric orbital integrals.
We still fix a maximal torus T of K with Lie algebra t. For simplicity, if γ ∈ G is semisimple, we may and we will assume that (6.0.2) γ = e a k, k ∈ T, a ∈ p, Ad(k −1 )a = a.
In this case, Recall that Z(γ e ) 0 is the identity component of the centralizer of γ e in G. Then The Cartan involution θ preserves Z(γ e ) 0 such that Z(γ e ) 0 is a connected linear reductive Lie group. Then we have the following diffeomorphism It is clear that δ(Z(γ e ) 0 ) = δ(G). Moreover, H is still a maximally compact Cartan subgroup of Z(γ e ) 0 .
Recall that T U is a maximal torus of U with Lie algebra t U = √ −1b ⊕ t ⊂ u. Let R + (u, t U ) be a positive root system for R(u, t U ), which is not necessarily the same as in Subsection 5.3 when δ(G) = 1.
Since U is the compact form of G, then U (γ e ) 0 is the compact form for Z(γ e ) 0 . Moreover, T U is also a maximal torus of U (γ e ) 0 . Let R(u(γ e ), t U ) be the corresponding real root system with the positive root system R + (u(γ e ), t U ) = R(u(γ e ), t U ) ∩ R + (u, t U ). Let ρ u , ρ u(γe) be the corresponding half sums of positive roots.
Let U (γ e ) be a connected finite covering group of U (γ e ) 0 such that ρ u , ρ u(γe) are analytically integrable with respect to the maximal torus T U of U (γ e ) associated with t U . It always exists by a similar construction as in the proof to Theorem 5.4.4.
Let K(γ e ) be the analytic subgroup of U (γ e ) associated with Lie algebra k(γ e ). By [20,Proposition 7.12], U (γ e ) has a unique complexification U (γ e ) C which is a connected linear reductive Lie group. Let Z(γ e ) be the analytic subgroup of U (γ e ) C associated with z(γ e ) ⊂ u(γ e ) C = z(γ e ) C . Then we have the following Cartan decomposition (6.0.6) Z(γ e ) = K(γ e ) exp(p(γ e )).
We still denote by θ the corresponding Cartan involution on Z(γ e ). The Lie group Z(γ e ) is a finite covering group of Z(γ e ) 0 . Moreover, we have the identification of symmetric spaces (6.0.7) X(γ e ) ≃ Z(γ e )/ K(γ e ).
Note that even under an additional assumption that G has compact center, Z(γ e ) may still have noncompact center. Let λ be a dominant weight for (U, T U ) with respect to R + (u, t U ). Let (E λ , ρ E λ ) be the irreducible unitary representation of U with highest weight λ. As before, let (F λ , ∇ F λ ,f , h F λ ) be the corresponding homogeneous flat vector bundle on X with flat connection ∇ F λ ,f . Let D X,F λ ,2 denote the associated de Rham-Hodge Laplacian.
Let W 1 U (γ e ) ⊂ W (u C , t U,C ) be the set defined as in (5.4.13) but with respect to the group U and to γ e = k ∈ T ⊂ T U . As in Definition 5.4.3, for σ ∈ W 1 U (γ e ), set .
We also view γ h = e a as a hyperbolic element in Z(γ e ). For σ ∈ W 1 U (γ e ), as in (6.0.1), we set Note that we use B| z(γe) on z(γ e ) to define this orbital integral for Z(γ e ). Set In particular, c(γ e ) = 1.
The following theorem is essentially a consequence of the generalized Kirillov formula in Theorem 5.4.4. Theorem 6.0.1. Let γ ∈ G be given as in (6.0.2). For t > 0, we have the following identity, We call (6.0.11) a localization formula for the geometric orbital integral.
(2) A similar consideration can be made for Tr s [γ] exp(−tD X,F λ ,2 ) , where (6.0.11) will become an analogue of the index theorem for orbifolds as in (2.2.11). The related computation can be found in [7,Subsection 10.4].

Full asymptotics of elliptic orbital integrals
In this section, we always assume that δ(G) = 1 and that U is compact. We also use the notation and settings as in Subsections 5.1, 5.2 and 5.3.
In this section, we are concerned with a sequence of flat vector bundles {F d } d∈N on X defined by a nondegenerate dominant weight λ. For elliptic γ, we will compute explicitly E X,γ (F d , t) and its Mellin transform in terms of the root data.
Note that when γ = 1, E X,γ (F d , t) is already computed by Müller-Pfaff [30] using the Plancherel formula for identity orbital integral. We here give a different approach via Bismut's formula as in (4.1.17), which inspires an analogue computation for any elliptic element. 7.1. A family of representations of G. Recall that T is a maximal torus of K, and T U is a maximal torus of U . Let W (U, T U ) denote the (analytic) Weyl group of (U, T U ), so that W (U, T U ) = W (u C , t U,C ). The positive root system R + (u, t U ) is given in Subsection 5.3. Recall that P ++ (U ) is the set of dominant weights of (U, T U ) with respect to R + (u, t U ).
Definition 7.1.1. A dominant weight λ ∈ P ++ (U ) is said to be nondegenerate with respect to the Cartan involution θ if It is equivalent to Note that if such dominant weight exists, we must have δ(G) > 0.
Let (E λ , ρ E λ ) be the irreducible unitary representation of U with highest weight λ. By the unitary trick, it extends to an irreducible representation of G, which we still denote by (E λ , ρ E λ ). Then λ being nondegenerate is equivalent to say that as G-representation (as in [30]).
Recall that a 1 ∈ b is such that B(a 1 , a 1 ) = 1.
Put F d = G × K E d . Let D X,F d ,2 denote the associated de Rham-Hodge Laplacian. For t > 0, let exp(−tD X,F d ,2 /2) denote the heat operator associated with D X,F d ,2 /2.
For t > 0, d ∈ N, if γ ∈ G is semisimple, as in (6.0.1), set It is clear that E X,γ (F d , t) only depends on the conjugacy class [γ] in G. If γ = 1, we also write

7.2.
Estimates for t small. By (4.1.17), (6.0.15), (6.0.16), if γ = k ∈ K, we have As t → 0, E X,γ (E d , t) has the asymptotic expansion in the form of Where a γ j (d) are functions in d.
Proof. If γ is elliptic, up to a conjugation, we assume that γ = k ∈ T . Thus H is also a Cartan subgroup of Z(γ) 0 , then b(γ) = b. Let b ⊥ (γ) be the orthogonal complementary space of b(γ) in p(γ), whose dimension is p − 1.
We now include a direct proof to (7.2.2) in general case. By (7.2.1), we have where the integral is rescaled by √ t. In this proof, we denote by C or c a positive constant independent of the variables t and Y k 0 . We use the symbol O ind to denote the big-O convention which does not depend on t and Y k 0 . The same computations as in [23,Eqs. (7.4.8) -(7.4.10)] shows that for Y k 0 ∈ k(k), It is clear that Combining (7.2.5) and (7.2.6), we see that there exists a number N ∈ N big enough, if t ∈]0, 1] The second estimate in (7.2.2) can be proved using the same arguments as in [23, eqs. (7.4.24) - (7.4.29)].
The asymptotic expansion in (7.2.3) is just a consequence of (7.2.2) and (7.2.4). This completes the proof of our proposition. 7.3. Identity orbital integrals for Hodge Laplacians. In this subsection, we compute I X (F d , t) using Bismut's formula (7.2.1). In next subsection, we connect our computational results to the ones obtained by Müller-Pfaff [30]. We will give in detail the main points in the computation, which are also applicable for computing E X,γ (F d , t) with any elliptic γ.
Let Vol(K/T ), Vol(U M /T ) be the Riemannian volumes of K/T , U M /T with respect to the restriction of −B to k, u m respectively. We have explicit formulae for them in terms of root data, for example, (2l)! is of degree strictly smaller than 4l.
We use the notation in Subsection 5.3. In particular, the positive root systems R + (u, t U ) and R + (u m , t) are fixed in Subsection 5.3. The set W u ⊂ W (U, T U ) is given by (5.3.6). As in Proposition 5.3.2, for ω ∈ W u V dλ+λ0,ω is an irreducible unitary representation of U M with highest weight η ω (dλ + λ 0 ) given by (5.3.11). Definition 7.3.1. For j = 0, 1, · · · , l, ω ∈ W u , set , Ω um 2l−2j max . Since dim V dλ+λ0,ω is a polynomial in d by the Weyl dimension formula, then Recall that for ω ∈ W u , a λ,ω , b λ0,ω are defined in Definition 7.1.2.
For t > 0, we have the following identity Remark 7.3.3. The formula (7.3.7) is compatible with the estimate (7.2.2). For example, if we take the asymptotic expansion of the right-hand side of (7.3.7) as t → 0, the coefficient of t −l−1/2 is given by Then by (5.3.10), if l ≥ 1, we get By (7.3.5) and (7.3.9), the quantity in (7.3.8) is 0 (provided l ≥ 1).
Before proving Theorem 7.3.2, we need some preparation work. Note that R(u m , t) contains R(k m , t) as a sub root system. Since the adjoint action of t preserves the splitting u m = √ −1p m ⊕ k m , then we can write where the two subsets are disjoint and R( √ −1p m , t) is the set of roots associated with the adjoint action of t on p m .
We have fixed a positive root system R + (u m , t) in Subsection 5.3, which induces a positive root system R + (k m , t) ⊂ R(k m , t). We also put For y ∈ t, put (7.3.14) We can always extend analytically the above functions to y ∈ t C .
It is clear that if y ∈ t C , π um/t (y) = π √ −1pm/t (y)π km/t (y), σ um/t (y) = σ √ −1pm/t (y)σ km/t (y) Recall that t U is a Cartan subalgebra of u. Then using R + (u, t U ) defined in Subsection 5.3, we can define the associated functions π u/tU (y), σ u/tU (y) for y ∈ t U as in Definition 7.3.4. Let R + (k, t) be a positive root system of R(k, t) such that it induces the same R + (k m , t). Then we define the functions π k/t (y), σ k/t (y) as in Definition 7.3.4. Recall that if Y k 0 ∈ k, we have Then by (7.2.1), we have By Weyl integration formula, we have )]e −|y| 2 /2t dy.
Moreover, we have the following identity Proof. Recall that as K M -modules, we have By (5.1.5) and (7.3.22), we get the following identification of K M -modules, Then if y ∈ t, Note that Then by (7.3.24), we get The proof to (7.3.21) is similar to the proof to [37,]. We include the details as follows. By (5.1.5) and (7.3.22), using [3,Eq.(7.5.24)], if y ∈ t, we have Similar for A(iad(y)| um ). Moreover, we can verify directly that if y ∈ t, (7.3.29) π k/t (iy) 2 = (−1) l π km/t (iy) 2 det(iad(y))| n C . By (7.3.16) and (7.3.28), if y ∈ t, Then by (5.1.5), (7.3.15), (7.3.23) and (7.3.29), for y ∈ t, we get If y ∈ t is such that π um/t (y) = 0, Recall that m = dim p. Then )]e −|y| 2 /2t dy. Note that the function in y ∈ t can be extended directly to an U M -invariant function in y ∈ u m . Since t is a Cartan subalgebra of u m , we can apply the Weyl integration formula again for the pair (u m , t), then we rewrite (7.3.34) as 2t dy. If y ∈ u m , then If y ∈ u m , by [37,], we have Note that (7.3.39) m + n = dim u m + 4l + 1.
Recall that Vol L (O ηω (dλ+λ0)+ρu m ) is the volume of O ηω (dλ+λ0)+ρu m with respect to the Liouville measure. By the Kirillov formula, we have We claim the following identity, Indeed, by (5.4.6), we have the following identity as elements in Λ · (u ⊥ (b) * ), Since U M acts on u ⊥ (b) isometrically with respect to −B, then Then (7.3.48) follows from (7.3.47) and from (7.3.49) -(7.3.51). The right-hand side of (7.3.48) is a polynomial in d and in t −1 . Recall that dim u ⊥ (b) = 4l. Then we can rewrite the right-hand side of (7.3.48) as follows, , Ω um 2l−2j max Then (7.3.7) follows from (7.3.4), (7.3.52). This completes the proof.
The Mellin transform of I X (F d , t) (if applicable) is defined by the following formula as a function in s ∈ C, Theorem 7.3.6. Suppose that λ is nondegenerate with respect to θ. For d ∈ N large enough and for s ∈ C with ℜ(s) ≫ 0 , MI X (F d , s) is well-defined and holomorphic, which admits a meromorphic extension to s ∈ C.
then we have The quantity PI X (F d ) is a polynomial in d for d large enough, whose coefficients depend only on the root data and λ, λ 0 .
Proof. Since λ is nondegenerate, by Lemma 7.1.3, a λ,ω = 0, ω ∈ W u . Then there Then we see that PI X (F d ) is a polynomial in d for d large enough. This completes the proof of our theorem.
As explained in Remark 5.3.3, when G has noncompact center with δ(G) = 1 (but U is still assumed to be compact), most of the above computations can be reduce into very simple ones. Recall that a λ , b λ0 ∈ R are defined in Definition 7.1.2. When λ is nondegenerate, a λ = 0. Corollary 7.3.7. Assume that U is compact and that G has noncompact center with δ(G) = 1, and assume that λ is nondegenerate. Then for t > 0, s ∈ C, Furthermore, Proof. By the hypothesis, we get that l = 0, W u = {1} and Q λ,λ0 0,1 (d) = dim E d . Then (7.3.57), (7.3.58) are just special cases of (7.3.7), (7.3.54) and (7.3.56).
However, we can prove them more directly using a result of Proposition 4.1.6. It is enough to prove the first identity in (7.3.57). Note that by (5.3.14), we have Combing (4.1.32) and (7.3.60) -(7.3.62), we get the first identity in (7.3.57), and hence the other identities. This gives a second proof to this corollary. 7.4. Connection to Müller-Pfaff 's results. In this subsection, we assume that G has compact center with δ(G) = 1. We explain here how to connect our computations in previous subsection to Müller-Pfaff's results in [30].
Let a 1 ∈ b * be which takes value −1 at a 1 .
Definition 7.4.1. For ω ∈ W u and Λ ∈ P ++ (U ), for z ∈ C, set Since θ fix Ω u(b) , by the fact that det θ| u ⊥ (b) = 1, we get that P ω,Λ (z) is an even polynomial in z.
As a consequence of (7.3.56) and Lemma 7.4.2, we have the following result.
where R(d) is a polynomial whose degree is no greater than the degree of dim E d .
An important step in Müller-Pfaff's proof to Proposition 7.4.4 is reducing the computation of PI X (F d ) to the cases where G = SL 3 (R) or SO 0 (p, q) with pq > 1 odd. Such reduction is already explained in Subsection 4.2. More precisely, we have (7.4.10) where X 1 is one case listed in (4.2.1), and X 2 is a symmetric space rank 0. We use the notation in Subsection 4.2. Let λ i , λ 0,i be dominant weights of U i , i = 1, 2 such that (7.4.11) λ = λ 1 + λ 2 , λ 0 = λ 0,1 + λ 0,2 .
Now we consider the sequence dλ + λ 0 , d ∈ N. Then Since G 2 is equal rank, the nondegeneracy of λ with respect to θ is equivalent to the nondegeneracy of λ 1 with respect to θ 1 . Then by Proposition 4.2.1, after taking the Mellin transform, we have Then we only need to evaluate PI X1 (F dλ1+λ0,1 ) explicitly, which has been done in [30,Section 6].

Asymptotic elliptic orbital integrals.
Definition 7.5.1. A function f (d) in d is called a pseudopolynomial in d if it is a finite sum of the term c j,s e 2π √ −1sd d j with j ∈ N, s ∈ R, c j,s ∈ C. The largest j ≥ 0 such that c j,s = 0 in f (d) is called the degree of f (d).
We say that the oscillating term e 2π √ −1sd is nice if s ∈ Q. We say that a pseudopolynomial f (d) in d is nice if all its oscillating terms are nice.
Remark 7.5.2. If f (d) is a nice pseudopolynomial in d, then there exists a N 0 ∈ N >0 such that the function f (dN 0 ) is a polynomial in d.
We will use the same notation as in Section 6. The following theorem is a consequence of the geometric localization formula obtained in Theorem 6.0.1.
Theorem 7.5.3. Suppose that γ ∈ G is elliptic and that λ is nondegenerate with respect to θ. If s ∈ C is with ℜ(s) ≫ 0, the Mellin transform ME X,γ (F d , s) of E X,γ (F d , t), t > 0 is well-defined and holomorphic. It admits a meromorphic extension to s ∈ C which is holomorphic at s = 0. Set (7.5.1) PE X,γ (F d ) = ∂ ∂s | s=0 ME X,γ (F d , s).
Then PE X,γ (F d ) is a pseudopolynomial in d (for d large). If γ is of finite order, then PE X,γ (F d ) is a nice pseudopolynomial in d.
More precisely, let k ∈ T be an element conjugate to γ in G. Let W 1 U (k) ⊂ W (U, T U ) be defined as in (5.4.13) with respect to R + (u, t U ). Then for σ ∈ W 1 U (k), σλ ∈ P ++ ( U (k)) is nondegenerate with respect to the Cartan involution θ on z(k). For d ∈ N, let E k σ,d be the irreducible unitary representation of U (k) with highest weight dσλ + σ(λ 0 + ρ u ) − ρ u(k) . This way, we get a sequence of flat vector bundles {F k σ,d } d∈N on X(k). Then we have ϕ U k (σ, dλ + λ 0 )PI X(k) (F k σ,d ).
For σ ∈ W 1 U (k), the term ϕ U k (σ, dλ + λ 0 ) defined as in (6.0.8) is an oscillating term, which is nice if γ is of finite order.
Therefore, by (7.5.4), ϕ U k (σ, dλ + λ 0 ) is a nice oscillating term in d. The rest part follows from the fact each PI X(k) (F k σ,d ) is a polynomial in d, where we can use Theorem 7.3.6, Corollaries 7.3.7 & 7.4.3 to compute them. This completes the proof of our theorem.
If we write down each polynomial PI X(k) (F k σ,d ) by the formulae as in (7.3.50), then we can get an explicit formula for PE X,γ (F d ) in terms of root data and λ, λ 0 .
In the sequel, we give a different way to evaluate PE X,γ (F d ) inspired by the computations in Subsection 7.3. Let γ ∈ G be elliptic, after conjugation, we may assume that γ = k ∈ T . Then T is also a maximal torus for K(γ) 0 , and b(γ) = b.
For γ = k ∈ T , set Then c G (1) is just the constant c G defined in (7.3.2). We will use the same notation as in Subsections 5.3 & 5.4. In particular, W u is defined by (5.3.6) as a subset of W (u C , t U,C ), and W 1 (γ) is defined by (5.4.13) as a subset of W (u m,C , t C ). Now we extend Definition 7.4.1 for γ ∈ T .

A proof to Theorem 1.0.2
In this section, we give a complete proof to Theorem 1.0.2, then Theorem 1.0.1 follows as a consequence. We assume that G is a connected linear real reductive Lie group with δ(G) = 1 and compact center, so that U is a compact Lie group. 8.1. A lower bound for the Hodge Laplacian on X. We use the notation as in Subsection 4. Recall that e 1 , · · · , e m is an orthogonal basis of T X or p. Put Let C g,H,E be its action on E via ρ E . Then (8.1.2) C g,E = C g,H,E + C k,E .
Let ∆ H,X be the Bochner-Laplace operator on bundle Λ · (T * X) ⊗ F associated with the unitary connection ∇ Λ · (T * X)⊗F,u . Put where R F is the curvature of the unitary connection ∇ F on F . Then Θ(F ) is a self-adjoint section of End(Λ · (T * X) ⊗ F ), which is parallel with respect to ∇ Λ · (T * X)⊗F,u . By [6, eq.(8.39)], we have D X,F,2 = −∆ H,X + Θ(F ). Let Ω · c (X, F ) be the set of smooth sections of Λ · (T * X) × F on X with compact support. Let ·, · L2 be the L 2 scalar product on it. If s ∈ Ω · c (X, F ), we have (8.1.5) D X,F,2 s, s L2 ≥ Θ(F )s, s L2 .
Let ∆ H,X,i denote the Bochner-Laplace operator acting on Ω i (X, F ), and let p H,i t (x, x ′ ) be the kernel of exp(t∆ H,X,i /2) on X with respect to dx ′ . We will denote by p H,i t (g) ∈ End(Λ i (p * ) ⊗ E) its lift to G explained in Subsection 3.2. Let ∆ X 0 be the scalar Laplacian on X with the heat kernel p X,0 t . In Proposition 8.2.1, each elliptic γ ∈ Γ is of finite order, therefore PE X,γ (F d ) is a nice pseudopolynomial. Since T (Z, F d ) is always real number, then (8.2.1) still holds if we take the real part of PE X,γ (F d ) instead.