Generalized Bernstein–Reznikov integrals

Abstract

We find a closed formula for the triple integral on spheres in R 2n × R 2n × R 2n whose kernel is given by powers of the standard symplectic form. This gives a new proof to the Bernstein-Reznikov integral formula in the n = 1 case. Our method also applies for linear and conformal structures. Primary 42C05; Secondary 11F67 · 22E45 · 33C20 · 33C55 Mathematics Subject Classification (2000)

Invariant trilinear forms for spherical degenerate principal series of complex symplectic groups

“…The case of Lorentz groups SO • (d, 1) was studied separately in [Dei06] and [CKØP11]. Closed formulas for other generalized Bernstein-Reznikov integrals were obtained in [CKØP11] in relation with the representation theory of different families of groups, namely Sp(n, R) and GL(n, R). The method of [CKØP11] also allowed to recover the case of Lorentz groups from another point of view.…”

Geometric analysis on small unitary representations of GL(N,R)

“…The proof parallels that of [5,Proposition 2.3]. For h ∈ C ∞ (S 2n−1 ) δ , we define a homogeneous function h μ−n ∈ V ∞ −μ,δ by h μ−n (rξ ) := r μ−n h(ξ ) r > 0, ξ ∈ S 2n−1 .…”

A program for branching problems in the representation theory of real reductive groups

“…• (discretely decomposable case) branching problems may be purely algebraic and combinatorial ( [12,13,15,26,28,29,32,49,50,59]); • (continuous spectrum) branching problems may have analytic features [8,52,57,63]. (For example, some special cases of branching laws of unitary representations are equivalent to a Plancherel-type theorem for homogeneous spaces.…”