Short Time Behavior of Hermite Functions on Compact Lie Groups

✍ Scribed by Jeffrey J. Mitchell

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 269 KB
• Volume: 164
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1999.3404

✦ Synopsis

Let p t (x) be the (Gaussian) heat kernel on R 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 R n . Recent work of Gross (1993) andHijab (1994) has led to the study of a new class of functions on a general compact Lie group, G. In analogy with the R 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 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 L p (g, +), where 1 p< , and + is a Gaussian measure on g. Similar theorems are established when G is replaced by GÂK, where K is some closed, connected subgroup of G.