Cauchy–Szegö kernels for Hardy spaces on simple Lie groups

For any simple real group G possessing unitary highest weight representations one can define the Hardy space H(G) . This is a Hilbert space formed by holomorphic functions in a ‘non–commutative’ tube domain Γ satisfying a Hardy–type condition (Γ is the interior of a non– commutative complex semigroup Γ containing the group G ). The space H(G) is identified with the bi–invariant subspace of L(G) carrying the holomorphic discrete series. The reproducing kernel K of the Hardy space is called the Cauchy–Szegö kernel. The main result of the paper is an explicit calculation of this kernel for 3 families of groups G : the symplectic groups Sp(n,R) , the metaplectic groups Mp(n,R) , and the pseudo–unitary groups SU(p,q) . Introduction Let H be a Hilbert space whose elements f are holomorphic functions in a complex domain D 1 . Then one can associate to H a kernel function K : if 1 More exactly, one should assume that H is continuously embedded into O(D) , the space of all holomorphic functions, equipped with the topology of uniform convergence on compact subsets. ISSN 0940–2268 / $2.50 C © Heldermann Verlag 242 Olshanski f1, f2, . . . is an arbitrary orthonormal basis in H then K(z, w) = f1(z)f1(w) + f2(z)f2(w) + . . . , z, w ∈ D, and the definition does not depend on the choice of the basis. The kernel K possesses the following properties: K is holomorphic in z and anti–holomorphic in w ; K(z, w) = K(w, z); K is a positive definite kernel on D × D ; for any w ∈ D the function K(·, w) lies in H ; for any function f ∈ H and any w ∈ D f(w) = (f,K(·, w))H (the reproducing property); in particular, for any z, w ∈ D (K(·, w), K(·, z))H = K(z, w); finally, the initial Hilbert space H is uniquely determined by its kernel function K . Classical examples are the Bergman and Hardy (H ) spaces in the unit disc |z| < 1 and the (say, right) half–plane Re z > 0. For the Bergman space, the square of norm ‖f‖2 is obtained by integrating |f(z)|2 over the domain with respect to the Lebesgue measure, and the corresponding kernel function has the form (1− zw)−2 (the disc) or (z + w)−2 (the half–plane). For the Hardy space, ‖f‖2 is defined as the integral of |f(z)|2 over the boundary, and the corresponding kernel function is (1− zw)−1 (the disc) or (z + w)−1 (the half–plane). It is well–known that these examples can be generalized to multidimensional bounded symmetric domains and to tube domains R+iC (where C is an open convex cone in R ); further generalizations involve interpolation between the Bergman and Hardy cases, vector–valued holomorphic functions etc. Note that various Hilbert spaces of holomorphic functions and their kernel functions naturally arise in connection with unitary highest weight representations. (See, e.g., Faraut and Korányi [3], Inoue [10], Stein and Weiss [22], Vergne and Rossi [23].) The present paper deals with multidimensional complex domains of another kind, which may be viewed as ‘noncommutative’ tube domains. Let D be an irreducible bounded symmetric domain and G be a connected group, locally isomorphic to the automorphism group of D . Then G is (locally isomorphic to) one of the groups SU(p, q), Sp(n,C), SO∗(2n), SO0(2, n) or certain 2 exceptional groups of type E . Let GC be the complexification of G . Then in GC there exist closed subsemigroups Γ with nonempty interior Γ 2 The boundary value of f(z) must be defined in a suitable way. 3 To simplify the discussion we tacitly assume here that G is linear, i.e., admits a global complexification; however, this assumption is not essential. Olshanski 243 such that Γ ⊃ G and Γ 6= GC (in particular, Γ and Γ are invariant with respect to the two–sided action of the group G). The existence of such semigroups is closely related to the existence of proper closed invariant convex cones C in g , the Lie algebra of G : there is a bijective correspondence Γ↔ C between semigroups and cones, and each open semigroup Γ has the form G exp iC , where C is the interior of the cone C . (See Olshanski [19], Hilgert and Neeb [6].) Thus, Γ indeed looks as a ‘noncommutative’ tube domain with skeleton G . It turns out that for any semigroup Γ one can define a Hilbert space H(Γ) ⊂ O(Γ0), which is an analogue of the Hardy H space. The space H(Γ) admits a canonical isometric embedding into L(G) as a two–sided G invariant subspace. When Γ is the so–called minimal semigroup, H(Γ) just carries the holomorphic discrete series of G (in general case H(Γ) carries a part of the holomorphic discrete series). Let K(z, w) be the kernel function of the space H(Γ) (it is also called the Cauchy–Szegö kernel). Then the kernel K(z, g), where z ∈ Γ, g ∈ G , defines the orthoprojection L2(G)→ H(Γ). In particular, if the semigroup is minimal then this is the orthoprojection onto the holomorphic discrete series. The idea of Hardy spaces carrying the holomorphic discrete series is due to Gelfand and Gindikin [4]. A construction of the spaces H(Γ) was given in author’s paper [20]. Further works in this direction are Hilgert and Ólafsson [7], Hilgert, Ólafsson, and Ørsted [8]. A natural problem is to compute the Cauchy–Szegö kernel explicitly, at least for the minimal semigroups. To this end, one can try to start from the following presentation of the Cauchy–Szegö kernel: Assume the center of G is finite. Let λ range over the set of the highest weights of the holomorphic discrete series representations Tλ occuring in the decomposition of H(Γ) (recall that if Γ is minimal, the whole holomorphic discrete series occurs). Let fdimλ stand for the formal dimension of Tλ , let χλ denote the character of Tλ , which admits a canonical holomorphic extension to Γ , and let w 7→ w be the antilinear antiautomorphism of the semigroup that extends the antiautomorhism g 7→ g−1 of the group G . Then we have