Short Time Behavior of Hermite Functions on Compact Lie Groups

Letpt(x) be the (Gaussian) heat kernel on Rnat timet. The classical Hermite polynomials at timetmay be defined by a Rodriguez formula, given byHα(−x, t) pt(x)=αpt(x), whereαis a constant coefficient differential operator on Rn. 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 Rncase, these “Hermite functions” onGare obtained by the same formula, whereinpt(x) is now the heat kernel on the group, −xis replaced byx−1, andαis a right invariant differential operator. Let g be the Lie algebra ofG. Composing a Hermite function onGwith the exponential map produces a family of functions on g. We prove that these functions, scaled appropriately int, approach the classical Hermite polynomials at time 1 asttends to 0, both uniformly on compact subsets of g and inLp(g, μ), where 1⩽p<∞, andμis a Gaussian measure on g. Similar theorems are established whenGis replaced byG/K, whereKis some closed, connected subgroup ofG.