Short time behavior of Hermite functions on compact Lie groups

Let p(t)(x) be the (Gaussian) heat kernel on R-n at time t. The classical H ermite polynomials at time I may be defined by a Rodriguez formula, given b y H-alpha(-x, t) p(t)(x) =alpha p(t)(x), where alpha is a constant coeffici ent differential operator on R-n. Recent work of Gross (1993) and Hijab (19 94) has led to the study of a new class of functions on a general compact L ie group, G. In analogy with the R-n case, these "Hermite functions" on G a re obtained by the same formula, wherein p(t)(x) is now the heat kernel on the group, -x is replaced by x(-I), and alpha is a right invariant differen tial operator. Let g be the Lie algebra of G. Composing a Hermite function on G with the exponential map produces a family of functions on g. We prove that these functions, scaled appropriately in t, approach the classical He rmits polynomials at time 1 as t tends to 0, both uniformly on compact subs ets of g and in L-p(g, mu), where 1 less than or equal to p < infinity, and mu is a Gaussian measure on g. Similar theorems are established when G is replaced by G/K, where K is some closed, connected subgroup of G. (C) 1999 Academic Press.