IngentaConnect Short Time Behavior of Hermite Functions on Compact Lie Groups

Author: Gross J.L.J.

Source: Journal of Functional Analysis, Volume 164, Number 2, June 1999 , pp. 209-248(40)

Abstract:

Let p_t(x) be the (Gaussian) heat kernel on Ropf ⁿ at time t. The classical Hermite polynomials at time t may be defined by a Rodriguez formula, given by H(-x, t) p_t(x)=p_t(x), where is a constant coefficient differential operator on Ropf ⁿ. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new class of functions on a general compact Lie group, G. In analogy with the Ropf ⁿ case, these “Hermite functions” on G are obtained by the same formula, wherein p_t(x) is now the heat kernel on the group, -x is replaced by x^-1, and is a right invariant differential operator. Let gfr be the Lie algebra of G. Composing a Hermite function on G with the exponential map produces a family of functions on gfr . We prove that these functions, scaled appropriately in t, approach the classical Hermite polynomials at time 1 as t tends to 0, both uniformly on compact subsets of gfr and in L^p( gfr , ), where 1 les p< infin , and is a Gaussian measure on gfr . Similar theorems are established when G is replaced by G/K, where K is some closed, connected subgroup of G. Copyright 1999 Academic Press.

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