Short time behavior of Hermite functions on compact Lie groups.

摘要

Let {dollar}psb{lcub}t{rcub}(x){dollar} be the (Gaussian) heat kernel on {dollar}Rsp{lcub}n{rcub}{dollar} at time t. The classical Hermite polynomials at time t may be defined by a Rodriguez formula, given by {dollar}Hsb{lcub}alpha{rcub}({lcub}-{rcub}x,t)psb{lcub}t{rcub}(x)=alpha psb{lcub}t{rcub}(x),{dollar} where {dollar}alpha{dollar} is a constant coefficient differential operator on {dollar}Rsp{lcub}n{rcub}.{dollar} 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 {dollar}Rsp{lcub}n{rcub}{dollar} case, these "Hermite functions" on G are obtained by the same formula, wherein {dollar}psb{lcub}t{rcub}(x){dollar} is now the heat kernel on the group, {dollar}{lcub}-{rcub}x{dollar} is replaced by {dollar}xsp{lcub}-1{rcub},{dollar} and {dollar}alpha{dollar} is a right invariant differential 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 Hermite polynomials at time 1 as t tends to 0, both uniformly on compact subsets of g and in {dollar}Lsp{lcub}p{rcub}{dollar}(g, {dollar}dmu),{dollar} where {dollar}1leq p