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The diagonal distribution for the invariant measure of a unitary type symmetric space

Let U denote a simply connected compact Lie group, let K denote the fixed point set for an involutive automorphism of U, and let m denote the U-invariant probability measure on the symmetric space U/K. Consider the geodesic embedding U/K into U of Cartan. In this paper we compute the diagonal distribution for m, relative to a compatible triangular decomposition of G, the complexification of U. This boils down to a Duistermaat-Heckman exact stationary phase calculation, involving a Poisson structure on the dual symmetric space G_0/K discovered by Evens and Lu.

Suppose that K is a simply connected compact Lie group, and let G denote the complexification. Given a triangular decomposition g = n − ⊕ h ⊕ n + , a generic g ∈ K has a unique "LDU decomposition", g = lmau, where l ∈ N − (lower triangular), u ∈ N + (upper triangular), and ma ∈ H, with m ∈ H ∩ K (unitary) and a ∈ exp(h R = h ∩ ik) (positive). A formula of Harish-Chandra (essentially equivalent to the Weyl dimension formula) asserts that for λ ∈ h * where the integral is with respect to normalized Haar measure, the product is over positive complex roots, and 2δ is the sum of the positive complex roots (there are various ways in which the c-function arises, and this formula has many extensions and interpretations; see e.g. [H2], especially §5-6 of chapter IV). The purpose of this paper is to present a generalization of this formula, and some of the related geometry, in which K is replaced by a compact symmetric space. 1 We also consider one additional ingredient: a triangular decomposition of g, g = n − ⊕ h ⊕ n + , which is Θ-stable and for which t 0 = h ∩ k is maximal abelian in k, where Θ is the involution corresponding to the pair (U, K).
Given this triangular structure, a generic element of φ(U/K) can be written as g = lwa φ ml * Θ , where l, a φ , m are roughly as before, and w ∈ T (2) 0 (elements of order two); the possible w index the connected components of the set of generic elements. In the special case in which Θ is an inner automorphism, the generalization of (0.1) which we consider is of the form where, given w, the product is over positive roots α which are of noncompact type for the involution Ad(w)Θ, and |W (·)| denotes the order of the Weyl group. The plan of the paper is the following. In §1 we compute the intersections of the φ-images in (0.3) with the triangular decomposition of G. A notable qualitative fact is that just as the map U/K → G/G 0 in (0.3) is a homotopy equivalence, so also are the intersections with the triangular decomposition of G. Possibly everything in this section is known; it can certainly be generalized and packaged in various ways (the canonical source is [Wolf]).
In §2 the general formulation and a proof of (0.4) is presented. It turns out that, in the inner case for example, the w = 1 term in (0.4) equals where g 0 = lau is an Iwasawa decomposition in G, ψ(g 0 K) = g 0 g * 0 = la ψ l * is an LDU decomposition, and the integrals are with respect to a G 0 -invariant measure. It is remarkable, although not a surprise, that a(g 0 ) −2δ dV (g 0 K) is the volume element for a symplectic form having a momentum map log(a(g 0 K)). Hence (0.5) can be evaluated using (a noncompact version of) the Duistermaat-Heckman exact stationary phase method. The symplectic structure was discovered by Evens and Lu, in a general setting ( [EL]); the relevance of this structure was pointed out to me by Foth and Otto ([FO]), to whom I am grateful.
It is natural to consider the more general integral for g ∈ G 0 \G/U . This is an eigenfunction for G-invariant differential operators on G/U . This can also be evaluated exactly, by the same method. In [Pi1,2] I have discussed conjectural generalizations of (0.1) and (0.4) to loop spaces, and other kinds of infinite symmetric spaces. The localization argument applies in a heuristic way. In appendix A there is a proof of (0.1), involving an explicit factorization of the integral, which has elements that seem useful in the loop space context. Notation. ·, · will denote the Killing form for g. For an automorphism θ of g, we will often write θ(x) = x θ , and more briefly, Ad(g)(x) = x g . We will write x = x − + x h + x + for the triangular decomposition of x ∈ g, and x = x k + x p for the Cartan decomposition of x ∈ g 0 . §1. Symmetric Spaces and Triangular Decomposition.
Throughout the remainder of this paper, U will denote a simply connected compact Lie group, Θ will denote an involution of U , with fixed point set K, and X will denote the quotient, U/K. This implies that K is connected and X is simply connected (Theorem 8.2 of [H1]).
Corresponding to the diagram of groups in (0.1), there is a Lie algebra diagram where Θ, acting on the Lie algebra level and extended complex linearly to g, is +1 on k and −1 on p. We let (·) − * denote the Cartan involution for the pair (G, U ). The Cartan involution for the pair (G, G 0 ) is given by σ(g) = g − * Θ . Since * , Θ, σ, and (·) −1 commute, our practice of writing g Θ for Θ(g), etc, should not cause any confusion.
We have natural maps (1. 2) The vertical arrows (given by inclusion) are homotopy equivalences; more precisely, there are diffeomorphisms (polar or Cartan decompositions) in each case given by the formula (g, X) → gexp(X) (mod G 0 in the last case). In turn there are totally geodesic embeddings of symmetric spaces ( 1.4) where the symmetric space structures are derived from the Killing form. A group element of the form g = g 1 g −σ 1 satisfies the equation g * = g Θ (i.e. gg σ = 1); g * = g Θ implies that Ad(g)•σ is an antilinear involution; and if g = g 1 g −σ 1 , then Ad(g) • σ = Ad(g 1 ) • σ • Ad(g −1 1 ), hence σ and Ad(g) • σ are inner conjugate. These considerations lead to the following well-known (1.5) Proposition. (a) In terms of g ∈ G, where {·} 0 denotes the connected component containing the identity. (b) The connected components of {g −1 = g * = g Θ } are determined by the map which sends g to the inner conjugacy class of the involution η = Ad(g) • Θ, subject to the constraint that η equals Θ in Out(U ) = Ad(U )\Aut(U ). A similar statement applies to {g * = g Θ }, with σ and antilinear automorphisms of G in place of Θ and involutions of K.
Consider the map ψ : G → G : g → gg σ . If we use right translation to identify the tangent space at any point of G with g, the derivative at g is given by Now suppose gg σ = 1. Since Ad(g) • σ is an involution, the spectrum of Ad(g) • σ is fixed. Thus the dimension of the −1 eigenspace of Ad(g) • σ is constant on {gg σ = 1}. It follows that ψ has constant rank on the connected components of ψ −1 (1). Since ψ is an algebraic map, this implies that {g * = g Θ } is an embedded submanifold. A similar argument applies to the intersection with U .
The action is isometric (for the symmetric space structure). The constancy of the rank of ψ on connected components is equivalent to the statement that the dimension of the isotropy subgroup for the action of G is constant on connected components of {gg σ = 1} (in fact this dimension is the same on all components). Hence the action of G must be transitive on connected components. The same applies to the same action of U on {g ∈ U : gg Θ = 1}. This implies (a). For the first part of (b), note that in fact the map is a universal covering for each connected component (For the identity component this covering is understood more intellibly by identifying the total space with U/K: where q(g 1 K) = Ad(g 1 ) • Θ • Ad(g 1 ) −1 ; to obtain a similar picture for another component, we replace Θ by Ad(g) • Θ, for some g in the component).
The second part of (b) is similar (We could also note that the inclusion {g −1 = g * = g Θ } → {g * = g Θ } is a homotopy equivalence, since we know this is true for the identity component, and we are free to change Θ to Ad(g) • Θ; the fact that the π 0 's are the same is a reflection of the fact that classifying Θ's and classifying σ's are canonically isomorphic problems (see e.g. 2. of §6, chapter 10 of [H])).
Remarks (1.8). (a) I do not know of a uniform way to define an invariant for the class of an involution η as in (b). However it is a simple matter to produce an invariant in a case by case manner from the classification of symmetric spaces (see Table V of [H1]).
(b) The groups and maps in (1.4) exist for any automorphism Θ of K. However it seems that there is a linear characterization of the φ-images (up to connectedness issues), and G 0 is a real form, only in the symmetric case, Θ 2 = 1.
Fix a maximal abelian subalgebra t 0 ⊂ k. We then obtain Θ-stable Cartan subalgebras for g 0 , u, and g, respectively, where a 0 ⊂ p (see (6.60) of [Kn]). We let T 0 and T denote the maximal tori in K and U corresponding to t 0 and t, respectively.
Let ∆ denote the roots for h acting on g; ∆ ⊂ h * R , where h R = a 0 ⊕ it 0 . We choose a Weyl chamber C + which is Θ-stable (to prove that C + exists, we must show that it 0 , the +1 eigenspace of Θ acting on h R , contains a regular element of g; this is equivalent to the fact that h 0 in (1.8) is a Cartan subalgebra). Since σ = −(·) * Θ and (·) * is the identity on h R , σ(C + ) = −C + .
Given our choice of C + , we obtain a Θ-stable triangular decomposition g = n − ⊕h⊕n + , so that σ(n ± ) = n ∓ . Let N ± = exp(n ± ), H = exp(h), and B ± = HN ± . We also let W = W (G, T ) denote the Weyl group, At the group level we have the Birkhoff or triangular or LDU decomposition for G, When we intersect this decomposition with φ(G/G 0 ), and the other spaces in (b) of (1.5), we obtain various decompositions. We will first determine the structure of the pieces in the {g * = g Θ } case (thus we initially ignore connectedness issues).
(a) The intersection {g * = g Θ } ∩ Σ G w is nonempty if and only if there exists w ∈ w ⊂ N U (T ), such that w * Θ = w; w is unique modulo the action the stability subgroup is given by given by l, h, L → lL −1 wh(lL −1 ) * Θ is a diffeomorphism onto the connected component containing w. This component is homotopic to the torus exp({Ad(w −1 )Θ| t = −1}).
(e) In particular for w = 1, the map then this decomposition is unique, but we will not require this at the outset.
We have g = g * Θ if and only if where L = L − L + denotes the decomposition induced by the diffeomorphism Thus (1.13) holds if and only if and (wh) * Θ = wh, or, using the fact that h is real, Consider part (a). If g is in the intersection, then we have just seen that w must satisfy w * Θ = w. Conversely, given a unitary representative w for w satisfying w * Θ = w, the intersection contains w and hence is nonempty. This proves (a).
Part (b) is straightforward. Now consider (c). We first write g ∈ {g * = g Θ } ∩ Σ G w uniquely as lωu, where l ∈ N − ∩ wN − w −1 , ω = wh, and u ∈ N + . For the first part of (c) we must prove that we can relax the constraint on l to arrange for u = l * Θ . We can write ( 1.17) where L ∈ N − ∩ wN + w −1 is arbitrary. We must prove the existence of L such that The basic fact is that this equation has a unique solution L ∈ N − ∩ N +w satisfying ωL σ ω −1 = L −1 , namely L = (u σ l) 1/2 (square root has an unambiguous meaning in a simply connected nilpotent Lie group). To see this simply plug such an L into (1.18). We obtain the equation L 2 = u σ l. The fact that u σ l, and its square root, satisfy ωL σ ω −1 = L −1 follows from (1.15), and uniqueness of the square root.
As we remarked previously, the existence of a solution L proves that B − has open orbits. The rest of part (c) is relatively straightforward, using (1.16).
The uniqueness of the solution L, subject to the constraint we imposed, implies the first part of (d). The second statement in (d) follows routinely from the first part.
For part (e), to clarify the statement, observe that Now suppose that w = 1. In this case w is in the kernel of the homomorphism T → T : w → ww Θ , and this equals the subgroup generated by T (2) 0 and exp(ia 0 ). We can modify w by multiplying by something in the image of the homomorphism T → T : λ → λλ −Θ . This image is exp(ia 0 ). Therefore we can choose w ∈ T (2) 0 , but this choice is unique only modulo the intersection of T (2) 0 and exp(ia 0 ). This proves (e).
Example (1.20). In the group case, is trivial. Thus in this group case, the set of generic elements (considered in part (e)) is connected, as we already know.
(1.22) Notation. Given w as in (c) of (1.11), we let Σ Having understood the intersection of {g * = g Θ } with the triangular decomposition, we now want to specialize this to the identity component. In an abstract way this is answered by (1.5b) (and Remark (1.8a)). Concerning open orbits, we have the following (1.23) Proposition. Suppose that w ∈ N U (T ) satisfies w −Θ = w. The following are equivalent: Hence the open orbits can be parameterized by either In addition, the w 1 K are exactly the T 0 fixed points in U/K.
Proof of (1.23). Determining the possible (open) B − orbits in G/G 0 is equivalent to determining the possible (open) G 0 orbits in B − \G. Thus the equivalence of (a) and (b) follows from Theorem 4.6 and its Corollaries in [Wolf]. The other statements are obvious.
In general it apparently remains an open question to systematically obtain representatives for all B − orbits in G/G 0 , from the intrinsic point of view (see [WZ] for the Hermitian symmetric case). In this regard the nonintrinsic point of view of (1.11) seems to have some utility.
Let q : G → G/B + denote the quotient map. The map q applied to the decomposition (1.9) induces the (more conventional) triangular stratification for the flag space, . As a consequence, for the pieces of the induced decomposition for U , there are diffeomorphisms The main point of this section is now to describe the generalization of this to U/K → G/G 0 . We consider the Iwasawa decomposition for G, which we write as ( 1.27) where A = exp(h R ). There is an induced right action is surjective, and induces a diffeomorphism ( 1.32) is a homotopy equivalence; each is homotopic to the torus exp( ; for such a w, and choice of w 1 , the diffeomorphism in (a) simplifies to (1.33) Proof of (1.29). In proving the first part of (a), it is convenient to work with U/K instead of φ(U/K). Thus we consider a point in the intersection of U/K and the This immediately implies that u is in the T × G 0 -orbit of w 1 , and this proves surjectivity of the first map in (a). For the second part of (a), we first calculate the stabilizer for the action (1.28) at the point w 1 . Suppose that t ∈ T and g 0 ∈ G 0 satisfy tu(w 1 g 0 ) = w 1 . This is equivalent to 1 , then (1.34) holds with t = T (g w 1 0 ). Thus the stabilizer is isomorphic to is connected and solvable. The torus part is isomor- ). This implies (1.31).
Part (c) follows from (a).
We want to explain how this proposition is related to familiar facts in special cases. First consider the group case X = K. We have already explained why the generic set is connected; see Example (1.20). In the group case, (1.1) has the form Given an arbitrary triangular decomposition k C =ñ − ⊕h ⊕ñ + , we obtain a Θ-stable triangular decomposition for g by defining n ± =ñ ± ×ñ ± , and h =h ×h. The Iwasawa factorization (1.27) is equivalent to the two Iwasawa factorizations The map in (c) of (1.29) (using (·) * in place of inverse) is given by (1.38) So a φ = a −1 1 a − * 2 = (a 1 a 2 ) −1 . Thus in (1.29) we are using exp(h R )g 0 K ∈ exp(h R )\K C /K as coordinate, which is completely equivalent to using l 1 or l 2 ∈ N − . From this point of view, l 1 is a horocycle coordinate. Now suppose that U/K is Hermitian symmetric. In this case Θ is inner, P = K C B + is a parabolic subgroup of G, and the natural map i : U/K → G/P is a U -equivariant isomorphism. The natural map η : G 0 /K → G/P is an open holomorphic embedding, and the image is contained i((U/K) r ), the regular set (see ch VIII of [H1], especially Prop 7.14).
There is a commutative diagram where the top arrow u is a diffeomorphism, by (b) of (1.29), and the map I is defined in the following way: given g 0 K ∈ G 0 /K, we can write u(g 0 ) = exp(ix)k uniquely, where x ∈ p, exp(itx)K is a geodesic of minimal length joining the basepoint to u(g 0 )K, and k ∈ K; we set I(g 0 K) = (g 0 k −1 ) −1 K. To see that the diagram is commutative, note that because g 0 k −1 ∈ G 0 , (g 0 k −1 ) −1 = (g 0 k −1 ) * Θ = exp(ix)a * Θ l * Θ , and l * Θ ∈ N + ; thus I(g 0 K) equals u(g 0 ) mod P . Thus in the Hermitian symmetric case, Σ U/K 1 is the usual model of G 0 /K inside U/K, but the parameterization in (b) of (1.29) is related to the natural holomorphic map η in a clumsy way. §2. Diagonal Distribution Then Ad(w)Θ is an involution, and n − ⊕ h ⊕ n + is Ad(w)Θ-stable. We also suppose that w ∈ φ(U/K).
Given g ∈ Σ φ(U/K) w , we factor g as in (e) of (1.11), where the product is over pairs (α, Ad(w)Θ(α)) of positive complex roots which are not of compact type for Ad(w)Θ, and M = |N where the sum is over representatives w for the connected components of φ(U/K) ∩ Σ U 1 .
Note that it does not matter whether we take α or Ad(w)Θ(α) in the product (2.3), because δ and λ are fixed by Ad(w)Θ. In the case in which Θ is inner, i.e. a 0 = 0, all roots are either of compact or noncompact type. Hence in this case the product in (2.3) is over the noncompact type roots.
To prove (2.2) we will first note that we can reduce to the case w = 1 in (2.3). We will then need several Lemmas.
To see that it suffices to prove (2.3) in the case w = 1, observe that lef t translation by w ∈ T (2) 0 , which is an isometric map for the Riemannian structure of U , maps the w = 1 component to the w-component: ( 2.5) This map interchanges the canonical factorization from that relative to Θ to the one relative to Ad(w)Θ: if g has the unique decomposition g = lma φ l * Θ , then wg has the unique decomposition wg = l w wma φ (l w ) * Ad(w)Θ . Since a φ is unchanged, the integral is evaluated in the same way, except the meaning of the roots changes. We henceforth suppose w = 1. Consider the parameterization (2.6) from (1.33). Note that a φ = a(g 0 ) −1 a(g 0 ) τ ∈ exp(it 0 ).
(2.7) Lemma. We have Proof of (2.7). We will consider a slight reformulation of the problem. We identify U/K with φ(U/K). Let S denote the inverse image of Σ U/K 1 in U , with respect to the projection U → U/K. Consider the lift of Φ in (2.6). We must show that the Jacobian for the mapping Ψ, with respect to the Riemannian structures induced by the Killing form, is equal to a constant times a 2δ φ . To do this we identify ia 0 , g 0 ⊖ a 0 , and u with the tangent spaces to exp(ia 0 ), A 0 \G 0 and U , respectively, using the exponential map and right translation (we use right translation because A 0 appears on the left). Let P : g → u denote the projection with kernel n − ⊕ h R . We compute (2.10) The operator Ad(t −1 ) preserves u-volume, so it can be ignored. Write a = a 1 a 0 , relative to the decomposition A = exp(it 0 )A 0 . Since a 0 ∈ G 0 , Ad(a 0 ) will preserve g 0 -volume. Thus the determinant of (2.10) equals the determinant of the map where l ′ = a −1 1 la 1 ∈ N − and P 1 is P followed by the projection to u ⊖ ia 0 . Given x ∈ g, if x = x − + x h + x + is its triangular decomposition, then maps n − into itself, (2.11) is given by Thus the determinant of (2.10) is the same as the (real) determinant of the map Because of the unipotence of Ad(l ′ ), this is equal to (2.14) We will now show that the integral (2.8) can be computed using a Duistermaat-Heckman exact stationary phase calculation. The relevant Poisson structure was discovered by Evens and Lu in a very general setting ( [EL]). We will introduce this structure directly, but to understand why it is natural the reader will need to consult the original paper.
To do calculations we will use the isomorphism of vector bundles and we will use the Killing form to identify p * with p.
Consider the Ad(T 0 ) and Ad(A 0 )-stable decomposition of g as a direct sum of subalgebras: Let pr g 0 denote the projection g → g 0 along this decomposition. Given x ∈ g, with triangular decomposition x = x − + x 0 + x + , The Evens-Lu Poisson bivector is given by where Ω(g 0 ) : p → p is given by The operator Ω satisfies the equivariance condition To understand Ω, it is useful to consider the augmented operatorΩ : g 0 → g 0 given byΩ (2.20) Relative to the decomposition g 0 = k ⊕ p, This augmented operator can be factored as the composition of four operators where the first operator is given by where l ′ = a 0 la −1 0 , and the last equality depends upon the fact that conjugation by l ′ a 1 (g 0 ) maps n − into itself, and that n − terms disappear when we use (2.16).
Proof of (2.24). For (a) let X = x g 0 , Y = y g 0 , x, y ∈ p. Then This is clearly skew-symmetric in X and Y , because σ preserves the Killing form and it is complex antilinear. For the second part of (a) we refer to [EL] (or see §3 of [FO] for an exposition specific to this case). For (b), note that (2.23) implies the kernel of T is ia 0 . Thus (2.22) implies This, together with (2.21), implies (b). For (c), note that in (2.22) the first, second and fourth operators preserve volume determined by the Killing form. The determinant of T (relative to the Killing form volumes) is the same as the determinant of the operator on n + which maps x + to (x l ′ a 1 + ) + . This determinant equals α>0 a 2α 1 = a 4δ 1 (2.27) Thus the Pfaffian is a 2δ 1 .
By (b) the tangent directions in G 0 /K determining the symplectic leaves are given by [g 0 , x] such that x u ⊥ a 0 . This is clearly A 0 -invariant, because u(a 0 g 0 ) = u(g 0 ). Thus the left action of A 0 permutes the symplectic leaves. The symplectic form is given by the formula (2.28) This form does not in general descend to a form on the quotient A 0 \G 0 /K. However (c) of the preceding Lemma does imply that the volume form descends.
(2.29) Proposition. (a) The action of T 0 is Hamiltonian with momentum map This momentum map is proper, and it is semibounded. (b) The symplectic measure is (where the invariant measure is suitably normalized).
Proof of (2.29). Part (a) is proven in [FO] (Lemma 3.3, which in turn refers to a result of Van Den Ban). Part (b) follows from (c) of (2.24).
We can now apply the Duistermaat-Heckman exact stationary phase method, as generalized to noncompact manifolds in [PW]. For definiteness we will consider the symplectic leaf through the basepoint of G 0 /K.
We must first find the fixed points of the T 0 action. Suppose that g 0 K is fixed by T 0 . If we choose λ ∈ T 0 which generates a dense subgroup of T 0 , this is equivalent to g −1 0 λg 0 ∈ K. Since T 0 is maximal abelian in K, we can assume (by multiplying g 0 on the right by k ∈ K if necessary) that g −1 0 λg 0 ∈ T 0 . Since N G 0 (T 0 ) = N K (T 0 )exp(h 0 ), g 0 K = a 0 K for some a 0 ∈ A 0 . Thus each symplectic leaf has exactly one T 0 fixed point. Since we are considering the leaf through the basepoint, there is just one T 0 fixed point, the basepoint.
If X denotes the element of t 0 corresponding to δ + Λ (Λ = −iλ), then the Pfaffian of the infinitesimal action of X at the basepoint equals where the product is over pairs of positive roots (α, Θ(α)) which are not of compact type. The Duistermaat-Heckman formula now implies (2.3) in the case w = 1. This concludes the proof of (2.2).
We end this section with two brief remarks. First, it is interesting to consider the integral for g ∈ G, which generalizes (2.8). When this is well-defined, (1) this is a function of g ∈ G 0 \G/U , (2) this is a G 0 -invariant eigenfunction for G-invariant differential operators on G/U ; see Lemma 5.15 of ch2 section 5 of [H2] (one is usually interested in U -invariant eigenfunctions, i.e. spherical functions). To explicitly evaluate (2.31), first note that G 0 exp(it 0 )U = G (this is existence of polar decomposition for the nonRiemannian symmetric space G 0 \G). Thus we can suppose that g = a ∈ exp(it 0 ). In this case (2.31) is an integral over A 0 \G 0 /C K (a). One can define a Poisson structure on G 0 /C K (a), using the Evens-Lu method, as above (see §3 of [FO]). As in (2.29), the momentum map can be identified with log(a 1 (ga)), and the symplectic volume of a symplectic leaf can be identifed with the form a 1 (g 0 a) −2δ dV A 0 \G 0 /C K (a) , via the projection to the double coset space. The fixed points for the T 0 action are of the form a 0 wC K (a), where a 0 ∈ A 0 and w ∈ W (K, T 0 )/W (C K (a), T 0 ) (see Prop 4.3 of [FO]). Thus (2.31) equals {w} a 1 (wa) −2(δ+Λ) w (δ + Λ) w −1 , α where given w ∈ W (K, T 0 )/W (C K (a), T 0 ), the product is over (1) pairs of positive roots (α, Θ(α)) which are not of compact type (relative to Θ), and (2) positive compact type roots which vanish on C k (a).
The second remark is that there is a kind of "dual" Poisson structure, on all of U/K, which can be used so that the sum (2.4) has the structure of an exact stationary phase calculation. In the terminology of the paper [EL], in this section we used the Lagrangian splitting (2.15), to obtain a Poisson structure on G 0 /K; the "dual" is the (Iwasawa) Lagrangian splitting g = u ⊕ (h R + n − ), which induces a Poisson structure on U/K. This will hopefully be taken up elsewhere.
Appendix. Special Features of the Group Case.
In this appendix we will present a proof of (0.1), using facts about Bott-Samelson resolutions of Schubert varieties. One rationale for including this appendix is that many of the arguments are valid in the more general context of Kac-Moody Lie algebras and groups. Throughout this appendix, we will use the notation and basic results in [Kac].
We start with the following data: A is an irreducible symmetrizable generalized Cartan matrix; g = g(A) is the corresponding Kac-Moody Lie algebra, realized via its standard (Chevalley-Serre) presentation; g = n − ⊕ h ⊕ n + is the triangular decomposition; b = h ⊕ n + the upper Borel subalgebra; G = G(A) is the algebraic group associated to A by Kac-Peterson; H, N ± and B are the subgroups of G corresponding to h, n ± , and b, respectively; K is the "unitary form" of G; T = K∩H the maximal torus; and W = N K (T )/T ≃ N G (H)/H is the Weyl group.
A basic fact is that (G, B, N G (H)) with Weyl group W is an abstract Tits system. This yields a complete determination of all the (parabolic) subgroups between B and G. They are described as follows.
Let Φ be a fixed subset of the simple roots. The subgroup of W generated by the simple reflections corresponding to roots in Φ will be denoted by W (Φ). The parabolic subgroup corresponding to Φ, P = P (Φ), is given by P = BW (Φ)B.
Given w ∈ N K (T ), we will denote its image in W/W (Φ) byw. The basic structural features of G/P which we will need are the Birkhoff and Bruhat decompositions respectively, where the indexing set is W/W (Φ) in both cases. The strata Σw are infinite dimensional if g is infinite dimensional, while the cells Cw are always finite dimensional. Our main interest lies in the Schubert varietyCw, the closure of the cell. Fixw ∈ W/W (Φ). We choose a representative w ∈ N (T ) of minimal length n; for definiteness we will always take w of the form w = r n · · · r 1 (A.3) where r i = i α i ( 0 1 −1 0 ), and i α i : SL 2 → G is the canonical homomorphism of This result is essentially (5) of [Kac] together with Tits's theory. We will sketch a proof for completeness.
Proof of (A.4). Let ∆ + denote the positive roots, ∆ + (Φ) the positive roots which are combinations of elements from Φ. The "Lie algebra of P " is p = Σg −β ⊕ b where the sum is over β ∈ ∆ + (Φ); this is the Lie algebra of P in the sense that it is the subalgebra generated by the root spaces g γ for which exp : g γ → G is defined and have image contained in P . The subgroups exp(g γ ) generate P . We also let p − denote the subalgebra opposite p: p = g −γ , where the sum is over γ ∈ ∆ + \ ∆ + (Φ). The corresponding group will be denoted by P − .
The cell C w is the image of the map N + → G/P : u → uwP . The stability subgroup at wP is N + ∩ wP w −1 .
At the Lie algebra level we have the splitting The second summand equals where the sum is over roots β > 0 with w −1 β ∈ −(∆ + \ ∆ + (Φ)). These roots β are necessarily real, so that exp : n + w → N + w ⊆ N + is well-defined. For q ∈ Z + let N + q denote the subgroup corresponding to n + q = span{g β : height(β) ≥ q}. Then N + /N + q is a finite dimensional nilpotent Lie group, and it is also simply connected. By taking q sufficiently large and considering the splitting (A.5) modulo n + q , we conclude by finite dimensional considerations that each element in N + has a unique factorization n = n 1 n 2 , where n 1 ∈ N + w and n 2 ∈ N + ∩ wP w −1 : The important point here is that modulo N + q we can control N + ∩ wP w −1 by the exponential map.
The inverse image of Cw under φ is SL ′ 2 × · · · × SL ′ 2 . Proof of (A.18). Let σ = r n−1 · · · r 1 . It suffices to show that for the natural actions and The first line, (A.19), follows from (A.4) since i α n (SL ′ 2 ) ⊆ exp(−g −α n )B and B × Cσ ⊆ Cσ. The second line follows from For the third line it's clear that the image of the left hand side is a union of cells, since we can replace r n i α n (SL 2 ) by P n . This image is at most n − 1 dimensional. Therefore it must have null intersection with Cw.
Fix an integral functional λ ∈ h * which is antidominant. Denote the (algebraic) lowest weight module corresponding to λ by L(λ), and a lowest weight vector by σ λ . Let Φ denote the simple roots α for which λ(h α ) = 0, where h α is the coroot, P = P (Φ) the corresponding parabolic subgroup. The Borel-Weil theorem in this context realizes L(λ) as the space of strongly regular functions on G satisfying for all g ∈ G and p ∈ P , where we have implicitly identified λ with the character of P given by λ(u 1 w exp(x)u 2 ) = exp λ(x) (A.24) for x ∈ h, u 1 , u 2 ∈ N + , w ∈ W (Φ). Thus we can view L(λ) as a space of sections of the line bundle If g is of finite type, then L(λ) = H 0 (L λ ); if g is affine (and untwisted), then L(λ) consists of the holomorphic sections of finite energy, as in [PS]. Normalize σ λ by σ λ (1) = 1.
(A.26) Proposition. Letw ∈ W/W (Φ), and let w = r n · · · r 1 be a representative of minimal length n. The positive roots mapped to negative roots by w are given by where h τ is the coroot corresponding to τ . Then σ w λ (r n i α n (g n ) · · · r 1 i α 1 (g 1 )) = n 1 a λ j j where g = a b c d ∈ SL 2 .
Proof of (A.26). The claim about the τ j is easily derived from (2.5). None of these roots lie in ∆ + (Φ), by the same argument as follows (2.5). Thus each λ j > 0. It follows that Πa λ j j is nonzero precisely on the set SL ′ 2 × · · · × SL ′ 2 . Now σ w λ , viewed as a section of L λ → G/P , is nonzero precisely on the wtranslate of the largest strata, wΣ 0 = wP − P = (P − ) w wP. (A.27) We claim the intersection of this withCw is Cw. In one direction Cw = N + ∩ (P − ) w wP ⊆ (P − ) w wP (A.28) by (2.6). On the other hand (N + ∩ (P − ) w ) is a closed finite dimensional subgroup of (P − ) w . Since (P − ) w is topologically equivalent to wΣ 0 , the limit points of Cw must be in the complement of wΣ 0 . This establishes the other direction. It now follows from (A.4) that σ w λ is also nonzero precisely on SL ′ 2 × · · · × SL ′ 2 , viewed as a function of (g n , · · · , g 1 ).
(A.35) Proposition. Suppose that g is finite dimensional. Given g ∈ K such that gT ∈ Σ 1 , we can write g uniquely as g = lmau, where l ∈ N − , m ∈ T , a ∈ exp(h R ), and u ∈ N + . Then where the integral is with respect to the normalized Haar measure of K, and 2δ denotes the sum of the positive complex roots.