Daniel Bump, Maki Nakasuji
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 138(5) 1595-1605 2010年5月 査読有り
Let G = GL(r+1) over a nonarchimedean local field F. The Kashiwara crystal B(infinity) 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(infinity). Thus the Gindikin-Karpelevich formula evaluates the integral of the standard spherical vector in the induced model of a principal series representation as a product Pi(1 - q(-1)z(alpha))/(1 - z(alpha)) where z is the Langlands parameter and the product is over positive roots. This may also be expressed as a sum over B(infinity). The corresponding equivalence over a metaplectic cover of GL(r+1) is deduced by using Kashiwara's similarity of crystals.