ANNALES MATHÉMATIQUES BLAISE PASCAL Mladen Božičević Constant term in Harish-Chandra ’ s limit formula

Let GR be a real form of a complex semisimple Lie group G. Recall that Rossmann defined a Weyl group action on Lagrangian cycles supported on the conormal bundle of the flag variety of G. We compute the signed average of the Weyl group action on the characteristic cycle of the standard sheaf associated to an open GR-orbit on the flag variety. This result is applied to find the value of the constant term in Harish-Chandra’s limit formula for the delta function at zero.


Introduction
Let G R be a semisimple Lie group, g R the Lie algebra of G R , g the complexification of g R , h R a Cartan subalgebra, h the complexification of h R , and X the flag variety of g.A classical formula of Harish-Chandra [4] for the delta function at zero states that lim λ→0 α>0 ∂(α)m λ = cm {0} .
Here λ ∈ ih * R is regular, m λ resp.m {0} is the canonical measure on the coadjoint orbit G R • λ resp.{0}, and ∂(α) is the differential operator on h * defined by a positive root α.Furthermore, the constant c = 0 if and only if h R is a fundamental Cartan subalgebra.
In [8] Rossmann suggested a geometric approach to Harish-Chandra's formula, which was based on his results relating invariant eigendistributions on g R and homology classes of the conormal variety of G R -action on X, and on the properties of the coherent continuation representation of the Weyl group.Rossmann's argument does not give the exact value of the The author was partially supported by a grant from the Ministry of Science, Education and Sport of Croatia .Keywords: Flag variety, equivariant sheaf, characteristic cycle, coadjoint orbit, Liouville measure.Math.classification: 22E46, 22E30.constant c in case h R is fundamental.One of the main motivations for the present paper was to compute the non-zero c.It turns out that considerably more theory is needed to obtain this information.In fact, our results rely heavily on the work of Schmid and Vilonen [11], [12].In more details, the integral formula for the character associated to a G R -equivariant sheaf on X [12], Theorem 3.8, and a local expression of the character [12], Theorem 5.27 are two of the main ingredients in our analysis.The argument we use to prove Harish-Chandra's formula is quite standard: instead of measures on the coadjoint orbits one studies the asymptotic behaviour of their Fourier transforms.By the general philosophy that goes back to the work of Harish-Chandra, these Fourier transforms represent the characters of representations.In explicit terms, the Fourier transform of the canonical measure m λ , under the appropriate positivity assumption on the parameter λ, represents the character of an induced representation.It is interesting to point out that Harish-Chandra's formula does not follow from the information about the leading term in the asymptotic expansion of the character of this induced representation.Rather, one has to consider the signed average of the character of the induced representation over the Weyl group.It turns out that this virtual character is up to a constant term, which we compute explicitly, equal to the character of a finite dimensional representation.These facts are actually established as an identity between homology cycles supported in the conormal variety of G R -action on X in Thorem 3.2, and the translation to the language of invariant eigendistributions is explained in Proposition 3.3 below.Harish-Chandra's formula is then deduced from the identity of homology cycles in Theorem 3.2, using the results in [12] and [8].When G R has a compact Cartan subgroup the character identity from Proposition 3.3 appears already without proof in [1].For this reason Proposition 3.3 can be considered as a generalization of a known result.Further applications of Schmid and Vilonen theory to the characters of Vogan-Zuckerman modules, and related limit formulas will be taken up in future publications.

Preliminaries
Suppose G R is a real, connected, linear, semisimple Lie group.We embed G R into a complexification G and denote by τ : G −→ G the involution on G having G R as the identity component of the set of fixed points.Next we choose a Cartan involution and extend it to G. Denote by K R resp.K the set of fixed points of θ on G R resp.G. Observe that θτ is a Cartan involution on G.We denote by U R the set of fixed points.Write g, k, g R , k R , u R for the Lie algebras of G, K, G R , K R , U R respectively.Denote the involutions on g induced by θ, τ by the same letters.In addition, let be the eigenspace decompositions defined by θ.Let ( , ) be the Killing form on g.We will use it whenever convenient to identify g and the dual space g * .
Next we introduce the notation related to the geometry of the flag variety.Write X for the flag variety of Borel subalgebras of g.
is independent on the choice of the pair (c, x), and is called the universal root system.Set . Given λ ∈ ∆, and a pair (c, x) as above, we write λ x = τ * x (λ).The universal Weyl group W is defined as the Weyl group of the root system ∆.Denote by ρ ∈ h * half the sum of the positive roots, and by h * the set of regular elements.Note that h resp.h * comes equipped with W -invariant symmetric bilinear form (•, •) whose specialization at x ∈ X coincides with the Killing form.In particular, if λ ∈ h * we write h λ for the element in h such that λ Let us recall the definition of the moment map and of the twisted moment map.Denote by T * X the cotangent bundle of the variety X.Given x ∈ X, denote by b ⊥ x ⊂ g * the space of linear forms vanishing on b x .We use the identification Denote by N * the cone of nilpotent elements in g * .Note that µ(T * X) = N * .The definition of the twisted moment map is due to Rossmann [9], §2.Note that any x ∈ X is fixed by a unique maximal torus C R ⊂ U R .We can use the decomposition g = c + [c, g] to view c * as a subspace of g * .Now we define the twisted moment map by By the result of Matsuki [7] G R acts on X with finitely many orbits.Moreover, G R -action on X is real algebraic, hence the orbits define a semialgebraic Whitney stratification of X.We shall denote by T * G R X union of the conormal bundles of the G R -orbits on X. Via the characteristic cycle map the K-group of G R -equivariant sheaves on X can be related to the top-dimensional homology group of the conormal variety of G Raction.In order to explain this, we need some additional notation.If Y is a locally compact space, we denote by H i (Y, C), i ∈ Z, the Borel-Moore homology groups with complex coefficients.Suppose that Y is a real algebraic manifold.The characteristic cycle CC(F) of a constructible sheaf F was defined by Kashiwara [5], Ch.IX, and [11].Recall that CC(F) is defined as a Lagrangian cycle in the real cotangent bundle T * Y .In fact, let S be a semi-algebraic Whitney stratification on Y, and F a sheaf constructible for S. Denote by T * S Y union of the conormal bundles to the strata.Then We remark that CC(F) is actually an integral homology cycle.However, in view of the applications we have in mind, it will be more convenient to consider complex coefficients.
Returning to the setting of the flag variety, we denote by Sh G R (X) the category of G R -equivariant constructible sheaves on X [2], Ch.0.Note that objects from Sh G R (X) are constructible for the orbit stratification on X.Let K(Sh G R (X)) be the Grothendieck group of the abelian category Sh G R (X).Since CC is additive on short exact sequences we obtain a homomorphism Let V be a coadjoint G-orbit in g * or a coadjoint G R -orbit in ig * R .To treat both cases simultaneously write V = G or V = G R , and denote by v the Lie algebra of V .The space In case V = G R , the form −iσ V is real valued, and we use the form (−iσ V ) k , 2k = dim R V to orient V. Then we define the measure m V by the formula and call it the Liouville measure.When V = G • λ, λ ∈ h * , we shall write ).Here a x and b x denote the tangent vectors which a, b ∈ u R induce by differentiation of the U R -action.
Denote by π X : T * X −→ X the natural projection, and by σ the canonical symplectic form on T * X.For λ ∈ h * the following formula holds [12], Proposition 3.3: Next we recall, following [12], § 3, the definition of invariant distributions on the Lie algebra as integrals of certain differential forms over the semialgebraic cycles in T * X.The Fourier transform of a test function φ ∈ C ∞ c (g R ) will be defined by without the usual i in the exponential.Here dx denotes a suitably normalized Lebesgue measure on g R .Let Γ be a semi-algebraic chain in T * X.
We say that converges and depends holomorphically on λ.In particular, this is true Denote by g R the set of regular semisimple elements in g R , and given a Cartan subalgebra c R ⊂ g R let c R = c R ∩ g R .By the work of Harish-Chandra Θ(CC(F), λ)|g R is a real analytic function.It is computed in [12] using the fixed point formalism of Goresky and MacPherson.Denote by X C R the set of fixed points of C R on X.Let ζ ∈ c R and x ∈ X C R .We select a subset ∆ x ⊂ ∆ + x closed under addition and having the property: We set further where RΓ N x (ζ) (•) stands for the local cohomology, and χ((•) x ) for the Euler characteristic of the stalk.Let F be an object from Sh G R (X).Then the restriction of Θ(CC(F), λ) to c R can be computed as follows [12], Theorem 5.27: We should point out that [12], Theorem 5.27 is deduced under the assumption that F is a (−λ − ρ)-monodromic sheaf.To prove (2.5) for a G R -equivariant sheaf, and arbitrary λ ∈ h * , one would have to argue similarly as in [12], §8-10.Alternatively, (2.5) follows from the main result in [6].Now we fix a θ-stable fundamental Cartan subalgebra c R ⊂ g R .Let be the Cartan decomposition, and c the complexification of c R .Next we recall the definition of the real Weyl group.Write On the other hand, we denote by W (g, c) the Weyl group of the root system ∆(g, c).Recall that W (g, c) is generated by the reflections s α , α ∈ ∆(g, c).
We consider W (g, c) also as a group of linear endomorphisms of c and c * .It is then not difficult to deduce Note that the involution θ acts naturally on ∆(g, c).Denote by the subset of roots vanishing on a.We choose positive root systems ∆ + 0 ⊂ ∆(g, c), and c).Denote by x 0 ∈ X, and x 1 ∈ X the points defined by the pairs (c, −∆ + 0 ) and (c, −∆ + 1 ) respectively.Define the orbits We conclude that the orbit S 0 is open in X.To the orbits S 0 and S 1 we associate standard sheaves Here, C S i denotes the trivial local system on S i , i = 0, 1.This notation for standard sheaves will be used in the rest of the paper.
Proposition 2.1.Let c R be a fundamental Cartan subalgebra, and ζ ∈ c R .Then Proof.Let x ∈ X C R .Since there are no real roots in the root system ∆(g, c) we can apply the same argument as in the proof of [12], Equation 7.37 to compute d ζ,x .We obtain We conclude Ad(g)c = c, hence gC R ∈ W (G R , C R ), as desired.

Weyl group modules
We begin the section be recalling some facts about Weyl group representations.When U ⊂ g * satisfies certain natural assumptions [9], Section 4.4 Rossmann defines a W -module structure on homology groups In particular, these assumptions are fulfilled in the following cases: , and ν ∈ N * .Note that in the first case we have Rossmann shows [9], Section 4.4, that inclusions of the orbit closures are compatible with W -module structure on homology groups.Moreover, if we set then we obtain an exhaustive filtration C) by W -submodules.Rossmann computes the corresponding graded module and shows that: In the case U = {ν} denote by C G (ν) the group of connected components of the centralizer of ν in G. Let d = dim C µ −1 (ν).Then C G (ν) acts on H 2d (µ −1 (ν), C) by permuting the irreducible components, and this action commutes with W -action. Hence ν) is irreducible [9], Theorem 4.5.This is the Springer representation associated to the orbit G • ν, and we denote the corresponding character by χ ν .If V is a W -module and χ an irreducible character of W we denote by Next we review briefly the definition of intertwining operators.Given a real algebraic manifold Y we denote by D(Y ) the bounded derived category of sheaves (of complex vector spaces) on Y constructible for semialgebraic stratifications.For w ∈ W denote by Y w ⊂ X × X the variety of pairs of Borel subalgebras in the relative position w, and by p 1 , p 2 : Y w −→ X projections onto the first and second factor in X × X.Then we define the intertwining functor attached to w by the formula: , One can show that I w is an equivalence of categories.Moreover, the equivalences I w induce an action of the Weyl group W on the K-group K(D(X)).
with dominant parameter, which results in a different sign in front of the orbital integral than in loc.cit.
Then the following formula holds Our goal is to study the asymptotic behaviour of the distribution Θ(CC(F 1 ), λ) when λ ∈ h * approaches zero.Some additional results are needed for this analysis.Denote by Θ O the Fourier transform of the Liouville measure m O .In more details Our computation will be based on the following simple formula: This formula is a special case of [8], Theorem 4.1.We remark that it can be also established by a straightforward calculation.The term o(λ n ) can be described as follows.By [10], Equation 6.16, we can find ∆ + 1 ⊂ ∆(g, c) satisfying the following conditions: −θ(∆ + 1 \ ∆(m, t)) = ∆ + 1 \ ∆(m, t), and ∆ + 1 ∩ ∆(m, t) = ∆ + ∩ ∆(m, t).Let x 1 ∈ X be the point determined by the pair (c, −∆ + 1 ).Recall that F 1 is the standard sheaf associated to the orbit S 1 = G R • x 1 .Set λ = τ * −1 where φ ∈ C ∞ (g R ).The formula from the theorem follows now by taking the inverse Fourier transform.
Let n = dim C X. Given x ∈ X we denote by b x the Borel subalgebra which fixes x , and by B x ⊂ G the Borel subgroup which stabilizes x via the adjoint action.Consider G-homogenous bundles B and [B, B] over X with fiber b x resp.[b x , b x ] at x ∈ X. Observe that B x acts trivially on b x /[b x , b x ], hence the G-bundle B/[B, B] is trivial.We denote by h its fiber, and call it the universal Cartan subalgebra.Note that h b x /[b x , b x ] canonically, for any x ∈ X.Let c ⊂ g be a Cartan subalgebra.Denote by ∆(g, c) the root system.Then c has |W | fixed points on X (| | stands for the cardinality), and we choose one of them: x ∈ X.Then c ⊂ b x , and we have a canonical isomorphism τ x : c → h.We denote by τ * sheaf on X. Write DF for the Verdier dual of F [5], Ch.III, and DF(x) for the restriction of DF to the open set N + x of X.Let E be the connected component of c R containing ζ. Finally we define the integers For any φ ∈ C ∞ c (g R ), o(λ n )(φ) is a holomorphic function of λ and lim t→0 o((tλ) n )(φ) t n = 0.Denote by C[h] resp.C[h *] the algebra of polynomial functions on h resp.h * .Write S(h) resp.S(h * ) for the symmetric algebra of h resp.h * .Recall that we have canonical isomorphismsC[h] ∼ = S(h * ) and C[h * ] ∼ = S(h).On the other hand the mapv → ∂(v), ∂(v)f (λ) = lim t→0 (f (λ + tv) − f (λ))/t, λ, v ∈ h * , f ∈ C ∞ (h * )extends to an isomorphism of S(h * ) and the algebra D(h * ) of differential operators on h * with constant coefficients.Thus we obtain an isomorphism of algebrasC[h] ∼ = D(h * ), p → ∂(p), p ∈ C[h].