Short time behavior of Hermite functions on compact Lie groups
Let \(p_{t}(x)\) be the (Gaussian) heat kernel on \(QTR{\mathbb R}^n\) at time \(t\). The classical Hermite polynomials at time \(t\) may be defined by a Rodrigues formula, given by \(H_{\alpha }(-x,t)p_{t}(x)=\alpha p_{t}(x)\), where \(\alpha \) is a constant coefficient differential operator on \(QTR{\mathbb R}^n\). 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 \(QTR{\mathbb R}^n\) 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 \(\alpha \) is a right invariant differential operator. Let \(QTR\mathfrak {g}\) be the Lie algebra of \(G\). Composing a Hermite function on \(G\) with the exponential map produces a family of functions on \(QTR\mathfrak {g}\). 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 \(QTR\mathfrak {g}\) and in \(L^{p}(QTR\mathfrak {g},\mu)\), where \(1\leq p<\infty \), and \(\mu \) is a Gaussian measure on \(QTR\mathfrak {g}\). Similar theorems are established when \(G\) is replaced by \(G/K\), where \(K\) is some closed, connected subgroup of \(G\).
constant coefficient differential operator
scientific article; zbMATH DE number 1309115
scientific article