Character Formulas and Localization of Integrals

CHARACTER FORMULAS AND LOCALIZATION OF INTEGRALS Wilfried Schmid In these notes, I shall describe two character formulas for semisimple Lie groups. Both are of interest by themselves, but the potential connections between the two formulas raise some intriguing questions. The formulas represent joint work with Kari Vilonen; full details will appear elsewhere. The formulas and their relation are well understood in the case of a compact group. For motivation, I shall start out with a discussion of the compact case. Thus I consider K, a connected, compact Lie group, and T K, a maximal torus. ⊂ I write kR, tR for the Lie algebras of K, T , and k, t for their complexified Lie algebras. Let π be an irreducible unitary representation of K. Because of the compactness of K, π must be finite dimensional. The Weyl character formula describes the character Θπ as a function of T , and thus – since the maximal torus meets every conjugacy class – globally as function on all of K. To recall the formula, I use the exponential map to identify the torus T with the quotient of its Lie algebra by the unit lattice, (1a) T tR/L , with L = unit lattice = x tR exp(x) = id . ≃ def { ∈ | } Dually, the character group T is isomorphic to the weight lattice, b ∗ (1b) T Λ = weight lattice = λ itR λ, L 2πiZ . ≃ def { ∈ | h i⊂ } b Under this isomorphism, λ Λ corresponds to eλ T , where eλ(exp x) = ehλ,xi ∈ ∈ ∗ for x tR. The Weyl group W = NK (T )/T operates on itR. This action preserves ∈ ∗ b a positive definite inner product ( , ) on itR. It also preserves the weight lattice and the root system Φ, i.e., the set· of· non-zero α Λ such that there exist x = 0 in k on which T acts according to the character eα.∈ The Weyl character formula6 can be stated as follows: ewλ (2) Θ = . π|T (1 e−α) wX∈W α∈Φ, (w(λ+ρ),α)>0 − Q Here λ = λ(π) is a particular element of the weight lattice, the highest weight of ∗ the representation π, and ρ a certain small vector in itR, independent of π, which will be described presently. The root hyperplanes µ itR (α, µ)=0 , α Φ, divide itR into a number of convex cones, the Weyl chambers{ ∈ ,| which are} permuted∈ simply transitively by W . Typeset by AMS-TEX 1 2 WILFRIED SCHMID Customarily, one singles out one of the Weyl chambers and calls it the dominant chamber. When K happens to be simply connected, ρ can be characterized as the shortest element of Λ lying in the interior of the dominant Weyl chamber. In general, the definition of ρ can be reduced to the case of a simply connected group: some finite covering of any connected, compact Lie group splits into the product of a torus and a simply connected compact group. To complete the statement of the Weyl character formula, one only needs to add that an element λ of the weight lattice Λ arises as highest weight of an irreducible representation precisely when it is dominant, i.e., when it lies in the (closed) dominant Weyl chamber. The Weyl character formula (2) is a beautiful, explicit formula – in some sense, one could hardly ask for more. Still, it has inspired attempts to explain it, or at least interpret it, in geometric terms. The Atiyah-Bott fixed point formula [1] provides one such geometric geometric interpretation. It depends on the Borel-Weil theorem [4,17], whose statement I need to recall. I fix a particular dominant weight λ. The splitting kR = tR [tR, kR] ∗ ⊕ induces a dual splitting of the vector space ikR, which allows me to think of λ and ∗ ρ as lying in ikR, and to define ∗ (3) Ωλ+ρ = K-orbit of λ + ρ in ikR . As homogeneous space, Ωλ+ρ is isomorphic to the quotient space K/T , and thus to the flag variety of k. Since λ exponentiates to a character eλ of the isotropy subgroup T of K at λ + ρ, it determines a K-equivariant line bundle (4) Ω . Lλ −→ λ+ρ Except for the multiplication by i = √ 1, Ωλ+ρ is a coadjoint orbit of K – an orbit in the dual of the Lie algebra; as such,− it carries a canonical symplectic structure. An complex polarization for Ωλ+ρ is a K-invariant pseudo-Kähler structure com- patible with the intrinsic symplectic structure. Complex polarizations exist; they correspond naturally and bijectively to the Weyl chambers. Once a polarization is chosen, the line bundle λ has a unique structure of K-equivariant holomorphic line bundle. The polarizationL corresponding to the dominant Weyl chamber is the unique positive polarization – positive in the sense that the pseudo-Kähler structure is positive definite, i.e., a true Kähler structure. Equip Ω with the positive polarization, and let ( ) denote the sheaf of λ+ρ O Lλ holomorphic sections of the line bundle λ. Since the action of K preserves the complex structures of the space Ω andL of the line bundle , it induces an λ+ρ Lλ action also on the cohomology groups of the sheaf ( λ). The Borel-Weil theorem describes the resulting representations of K : O L p H (Ωλ+ρ, ( λ)) = 0 if p =0 , (5) O L 6 H0(Ω , ( )) is non-zero, irreducible, and has highest weight λ , λ+ρ O Lλ provided λ is dominant and the polarization positive, as has been assumed. Bott’s generalized Borel-Weil theorem [4,11] describes the cohomology groups attached to an arbitrary invariant polarization and every λ Λ ; the precise statement need not concern us here. ∈ CHARACTER FORMULAS AND LOCALIZATION OF INTEGRALS 3 The maximal torus T operates on Ωλ+ρ with isolated fixed points, namely the various W -translates of the base point λ + ρ. The Atiyah-Bott fixed point formula, applied to the T -action on Ωλ+ρ and the holomorphic line bundle λ, expresses the p L alternating sum of the T -characters of the cohomology groups H (Ωλ+ρ, ( λ)) as a sum of terms corresponding to the fixed points; the term correspondingO L to any one of the fixed points w(λ + ρ), w W , turns out to be the summand in the formula (2) indexed by w. In this way,∈ the Weyl character formula becomes a formal consequence of the fixed point formula, as was observed already by Atiyah and Bott. Kirillov’s formula provides a totally different expression for the irreducible characters of K. By definition, 1/2 ∗ (6) θπ = (det exp∗) exp Θπ is the character on the Lie algebra of the irreducible representation π. Except 1/2 for the factor (det exp∗) – which has the effect of relating invariant differential operators on the group and on the Lie algebra – this is the pull-back of the global character Θπ to the Lie algebra. It is a smooth, bounded function, but is not square integrable. The exponential map of the connected, compact Lie group K is surjective and generically non-singular, so θπ completely determines Θπ. It will be ∞ convenient to define the Fourier transform φ of a test function φ Cc (kR) without the customary factor of i = √ 1 in the exponent, as a function∈ on ik∗ : − b R (7) φ(ζ) = φ(x)ehζ,xidx . ZkR b The Fourier transform of θπ exists as distribution, in the weak sense; with the convention (7), it is given by the relation (8) θπ = integration over Ωλ+ρ b (integration relative to the measure induced by the canonical symplectic structure). Kirillov calls this the “universal formula” for irreducible characters [10]; he estab- lished it not only in the compact case, but more importantly, in the nilpotent case where he used it to describe the unitary dual. One may picture the Weyl character formula, its interpretation in terms of fixed points, and Kirillov’s formula for θπ as the vertices of a triangle. One side of the triangle – the connection between the Weyl character formula and its fixed point b interpretation – is filled in by the Atiyah-Bott fixed point formula. Kirillov’s proof of (8) provides the second side, by direct computation. Berline-Vergne posed, and later answered the question of whether one can fill in the third side of the triangle: an identification of the right hand side of (8) with the sum of terms attached to the fixed points of the action of T on Ωλ+ρ , without the “detour” via the Weyl character formula. The argument of Berline-Vergne involves the K-equivariant cohomology groups ∗ C HK (Ωλ+ρ, ). These were defined, for any manifold with a differentiable K-action, 4 WILFRIED SCHMID by Cartan, who also expressed this cohomology in terms of a complex of differentiable forms, (9a) H∗ (Ω , C) H∗( ·(Ω ) S(k∗) K ) ; K λ+ρ ≃ {A λ+ρ ⊗ } · ∗ ∗ here (Ωλ+ρ) denotes the deRham complex, S(k ) the symmetric algebra of k , and AK · . the space of K-invariants, relative to the translation action on (Ωλ+ρ) and{ } the conjugation action on S(k∗). The differential A (9b) d : ·(Ω ) S(k∗) K ·(Ω ) S(k∗) K {A λ+ρ ⊗ } −→ {A λ+ρ ⊗ } is the sum of the deRham differential and a term involving the infinitesimal action of k as a Lie algebra of vector fields and the multiplication map k∗ S(k∗) S(k∗). It should be remarked that the equivariant cohomology can be defined⊗ whether→ or not K is compact. The description (9), on the other hand, depends crucially on the compactness of K.

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