Let G = GLr+1 over a nonarchimedean local field F . The Kashiwara crystal B(∞) is the quantized enveloping algebra of the lower triangular maximal unipotent subgroup N−. Examples are given where an integral over N−(F ) may be replaced by a sum over B(∞). Thus the Gindikin-Karpelevich formula evaluates the integral of the standard spherical vector in the induced model of a principal series repre...

For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic analogues of the twistor operators known from Riemannian spin geometry. We prove...

We study the Weyl representation of metaplectic operators suggested by earlier work of Mehlig and Wilkinson. We give precise calculations for the associated Maslov indices; these intervene in a crucial way in the Gutzwiller formula of semiclassical mechanics.

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg and Hoffstein [3] and Brubaker, Bump and Friedberg [4] provided n is sufficiently large; their coefficients involve n-th order Gauss sums. The case where n is small is harder, and is addressed in this paper when Φ = Ar. “Twisted” ...

The main purpose of this article is to supplement the authors’ results on degenerate principal series representations of real symplectic groups with the analogous results for metaplectic groups. The basic theme, as in the previous case, is that their structures are anticipated by certain natural subrepresentations constructed from Howe correspondence. This supplement is necessary as these repre...

Metaplectic quantum basis is a universal multi-qutrit quantum basis, formed by the ternary Clifford group and the axial reflection gate R = |0〉〈0| + |1〉〈1| − |2〉〈2|. It is arguably, a ternary basis with the simplest geometry. Recently Cui, Kliuchnikov, Wang and the Author have proposed a compilation algorithm to approximate any twolevel Householder reflection to precision ε by a metaplectic cir...

We give a classification of 1 st order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations , which are derived from the so called higher symplectic, harmonic or generalized Kostant spinor modules. Higher symplectic spinor modules are aris...