July 6, 1997

Characters, characteristic cycles and nilpotent orbits

This talk is a survey of (some parts of) joint work with Wilfried Schmid. These notes are in very rough from. For a more detailed account see the announcement [SV1]. Two thirds of this work is written up in full detail and is available at: http://www.math.harvard.edu/˜ schmid.

The Barbasch–Vogan conjecture. Let G_{ℝ} be a linear connected semisimple Lie group. We denote by G = (G_{ℝ})_{c} its complexification. Let K_{ℝ} ⊂ G_{ℝ} be a maximal compact subgroup which is the fixed points of a Cartan involution θ, and denote by K its complexification. The corresponding Lie algebras will be denoted 𝔤_{ℝ}, 𝔤, 𝔨_{ℝ}, 𝔨. Let N ⊂ 𝔤 ≅ 𝔤^{∗} be the nilpotent cone.

For an irreducible representation π of G_{ℝ}, two invariants were introduced some years ago. The first one is called the Asymptotic Support and was introduced in [BV]. Let Θ_{π} be the distribution character of π. The lift to the Lie algebra θ_{π} has an asymptotic expansion near zero. The leading term is of the form

(1)  ∑ a_{j}μ̂_{j}

where μ_{j} are the canonical measures of certain nilpotent orbits in N ∩ i𝔤_{ℝ} and a_{j} are constants in ℂ.

The second invariant was introduced in [V]. If V is the Harish–Chandra module attached to π, then choose a good K–invariant filtration ⋯ ⊂ V_{j} ⊂ V_{j+1} ⊂ … . Then the graded module gr_{.}(V) is an S(𝔤) module, so one can think of it as a K–equivariant coherent sheaf on 𝔤. It is supported on N ∩𝔭 (𝔤 = 𝔨+𝔭 is the Cartan decomposition) and its support is K-invariant, i.e., it is a union of K-orbits O′_{j} on N ∩ 𝔭. Generically a coherent sheaf is a vector bundle on its support. Hence we can associate to π a linear combination

(2)  ∑ b_{j}O′_{j}

called the Associated Cycle. In this formula the constants b_{j} are positive integers, they are the ranks of the above vector bundle on O′_{j}

It is natural to conjecture that the two expressions match via the Kostant– Sekiguchi correspondence, which is a bijection between nilpotent orbits on N ∩ 𝔭 and N ∩ i𝔤_{ℝ}. This bijection is as follows:

G_{ℝ}–orbits O_{j} ⊂ i𝔤_{ℝ} ∩ N are in 1–1 correspondence with θ–stable maps ϕ :

sl(2, ℂ) → 𝔤 with ϕ(sl(2, ℝ)) ∈ 𝔤_{ℝ} such that ϕ( \begin{matrix}0 & i \\ 0 & 0 \end{matrix} ) ∈ O_{j}. Then the orbit O′_{j} is

the K–orbit of ϕ( \begin{matrix}1 & i \\ i & −1 \end{matrix} ). The statement of the following theorem is what is known

as the Barbasch–Vogan conjecture.

Main Theorem. In formulas (1) and (2), O_{j} ↔ O′_{j} via the Kostant–Sekiguchi correspondence and a_{j} = b_{j}.

Typeset by AMS-TEX

Corollary. The constants a_{j} are non-negative integers.

The idea of the proof is to give a geometric interpretation of the Asymptotic Support and the Associated Cycle and then relate the two geometrically. We will outline the various steps of the proof.

We begin with the diagram (see [KSd]):

(3)

{G_{ℝ}–representations} ⟷ {Harish–Chandra modules}
↑  ↑

D_{G_{ℝ}}(X)_{−λ}  ⟷_{[MUV]}^{γ}  D_{K}(X)_{−λ}

Here X is the flag manifold, λ ∈ H^{q}(X, ℂ) ≅ 𝔥^{∗} determines an infinitesimal character, D_{G_{ℝ}}(X)_{−λ} is the derived category of G_{ℝ}–equivariant sheaves on X with twist −λ, and D_{K}(X)_{−λ} is the analogous category of K–equivariant sheaves. The upward arrows are given as follows:

(4)

D_{G_{ℝ}}(X)_{−λ} ⟶ {virtual representations of G_{ℝ}}
F ↦ χ(Ext^{∗}(𝔻F, O_{X}^{an}(λ)),

D_{K}(X)_{−λ} ⟶ {virtual Harish–Chandra modules}
F ↦ χ(Ext^{∗}(𝔻F, O_{X}^{alg}(λ))).

The map γ comes from [MUV]:

(5)  \begin{matrix}G_{ℝ} & ⟶ & G_{ℝ}/K_{ℝ} × X \\ a↓ & & p↓ \\ X & & X \end{matrix} .

The map a is a(g, x) = g^{−1} · x, and the map p is projection on the second factor. Given a sheaf F ∈ D_{K}(X)_{−λ}, there is a unique F̃ on G_{ℝ}/K_{ℝ} × X such that a^{!}F = p^{!}F̃. Then define γ(F) := Rp_{!}(F̃) ∈ D_{G_{ℝ}}(X)_{−λ}.

We embed (3) into a larger commutative diagram:

(6)

{G_{ℝ} representations} ⟷ {HC modules}
/  ↑  ↑  \
D_{G_{ℝ}}(X)_{−λ}  ⟷^{γ}  D_{K}(X)_{−λ}
BV  CC↓  ↓CC  V

\overline{H}_{top}(T_{G_{ℝ}}^{∗}X) Φ induced by γ ⟵ \overline{H}_{top}(T_{K}^{∗}X)

↘  grμ_{λ}^{∗}↓  ↓grμ_{λ}^{∗}  ↙
{cycles on i𝔤_{ℝ} ∩ N }

ϕ induced by Φ

⟷_{K−S}

{cycles on 𝔭 ∩ N }

The rest of the talk is devoted to explaining the terms in this diagram and indicating why the diagram commutes. We begin by explaining the notion of a characteristic cycle

(7)  CC : D_{c}^{b}(X) → {Lagrangian cycles on T^{∗}X},

due to Kashiwara (see, [KSa]). Here, temporarily, X denotes any (oriented) algebraic manifold and D_{c}^{b}(X) denotes the semi-algebraically constructible bounded derived category of sheaves on X. The following “axioms” uniquely determine, and allow one to geomterically compute CC(F):

(a) CC(ℂ_{X}) = [X].

(b) CC descends to the Grothendieck group of D_{c}^{b}(X)

(c) CC is locally defined

(d) if j : U → X is open, then

CC(Rj_{∗}F) = lim_{s→0^{+}} CC(F) + sd(log f),

where f is any defining equation of the boundary of U. For the fact that the last axiom holds in the real algebraic setting see [SV2].

Let us return to our situation and to the diagram (6). Let μ : T^{∗}X → N be the moment map. Then μ^{−1}(𝔭) = T_{K}^{∗}X the union of conormal bundles to the orbits of K on X. Similarly, μ^{−1}(i𝔤_{ℝ}) = T_{G_{ℝ}}^{∗}(X). Restricted to the categories D_{G_{ℝ}}(X)_{−λ} and D_{K}(X)_{−λ} the map CC takes values as indicated in diagram (6). The map grμ_{λ}^{∗} in diagram (6) is defined by integrating e^{λ} over the fibers of the map μ_{λ} and taking the “leading term”. To compute the map Φ induced by γ and the map ϕ induced by Φ we proceed as follows.

Let f : X → Y be proper proper map between algebraic manifolds. Then there is a diagram

(8)  \begin{matrix}T^{∗}X & ⟵^{df} & X ×_{Y} T^{∗}Y & ⟶^{ρ} & T^{∗}Y \\ ↓ & & ↓ & & ↓ \\ X & & X & ⟶ & Y \end{matrix}

Under these hypotheses Kashiwara-Schapira give the following formula:

(9)  CC(Rf_{∗}(F)) = ρ_{∗}df^{∗}CC(F).

The definition of Φ involves understanding the effect of the non-proper pushforward Rp_{!} on characteristic cycles. For this we embed G_{ℝ}/K_{ℝ} in its closure to make p proper and use (d) with the defining function e^{−<x,x>} for the boundary of G_{ℝ}/K_{ℝ}, here < , > denotes the Killing form. The function e^{−<x,x>} vanishes to infinite order on the boundary and hence is not real algebraic. Its use is justified by results of van den Dries-MacIntyre-Marker, see [DM] and [SV2].

We get the following formula for ϕ:

(10)  ϕ(C) = lim_{s→0^{+}}{sAd(exp(Rez))z | z ∈ C}.

Theorem. ϕ coincides with the Kostant–Sekiguchi correspondence.

It remains to show that the vertical arrows give the Asymptotic Support and the Associated Cycle, respectively. For the Associated Cycle this follows from a result of Chang [C]: One observes that the associated cycle is defined essentially in the same way as a characteristic cycle of a D-module. The argument for the Asymptotic Support is more subtle. One uses the following result. Let θ(F) be the character (on the Lie algebra) of the representation corresponding to the sheaf F ∈ D_{G_{ℝ}}(X)_{−λ}. For λ regular, there is a map μ_{λ} : T^{∗}X → Ω_{λ} = G · λ given by μ_{λ} = μ + v_{λ}. Here μ is the moment map from before. To define v_{λ} let U be a compact form of G which is compatible with K and G_{ℝ}. Choose a point x_{0} ∈ X. Then v_{λ} : X → Ω_{λ} is given by v_{λ}(ux_{0}) = u · λ. We have the following generalization of “Kirillov’s universal character formula”:

Theorem. For any ϕ ∈ C_{c}^{∞}(𝔤_{ℝ}),

θ(F)(ϕ) = \frac{1}{(2πi)^{n}} ∫_{μ_{λ}(CC(F))} ϕ̂σ_{λ}^{n}.

Remarks.

(1) The map μ_{λ} as well as the idea that such an integral formula should hold for some cycle, without specifying it to be μ_{λ}(CC(F)), is due to Rossman. Our contribution is proving the formula and specifying the cycle to be μ_{λ}(CC(F)).

(2) In the case when F corresponds to a tempered representation, the formula in the theorem is equivalent to Kirillov’s formula.

References

[BV] D.Barbasch and D.Vogan, The Local Structure of Characters, Jour. Func. Anal. 37 (1980), 27–55.

[C] J-T.Chang, Characteristic cycles of discrete series for ℝ-rank one groups, Trans. Amer. Math. Soc. 341 (1994), 603–622..

[DM] L.van den Dries and C.Miller, Geometric categories and o-minimal structures, Duke Math. J. 84, 497–540..

[KSa] M.Kashiwara and P.Schapira, Sheaves on manifolds, Springer, 1990.

[KSd] M.Kashiwara and W.Schmid, Quasi-equivariant D-modules, equivariant derived category, and representations of reductive Lie groups, Lie Theory and Geometry, in Honor of Bertram Kostant, Progress in Mathemtics, Birkhäuser, 1994, pp. 457–488.

[MUV] I.Mirković, T.Uzawa, and K.Vilonen, Matsuki Correspondence for Sheaves, Inventiones Math. 109 (1992), 231–245.

[SV1] W.Schmid and K.Vilonen, Characters, characteristic cycles, and nilpotent orbits, Geometry, Topology and Physics for Raoul Bott, International Press, 1995, pp. 329–340.

[SV2] W.Schmid and K.Vilonen, Characteristic cycles of constructible sheaves, Inventiones Math. 124 (1996), 451–502.

[V] D.Vogan, Gelfand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), 75–98.