Fourier and small ball estimates for word maps on unitary groups

Abstract

To a non-trivial word $w(x_1,...,x_r)$ in a free group $F_r$ on $r$ elements and a group $G$, one can associate the word map $w_G:G^r\to G$ that takes an $r$-tuple $(g_1,...,g_r)$ in $G^r$ to $w(g_1,...,g_r)$. If $G$ is compact, we further associate the word measure ${\tau }_{w,G}$, defined as the distribution of $w_G({\mathsf{X}}_1,...,{\mathsf{X}}_r)$, where ${\mathsf{X}}_1,...,{\mathsf{X}}_r$ are independent and Haar-random elements in $G$. In this paper we study word maps and word measures on the family of special unitary groups ${\left\{{\mathrm{SU}}_n\right\}}_{n\ge 2}$. Our first result is a small ball estimate for $w_{{\mathrm{SU}}_n}$. We show that for every $w\in F_r\setminus \left\{1\right\}$ there are $\epsilon (w),\delta (w)>0$ such that if $B\subseteq {\mathrm{SU}}_n$ is a ball of radius at most $\delta (w)\mathrm{diam}({\mathrm{SU}}_n)$ in the Hilbert-Schmidt metric, then ${\tau }_{w,{\mathrm{SU}}_n}(B)\le ({\mu }_{{\mathrm{SU}}_n}(B){)}^{\epsilon (w)}$, where ${\mu }_{{\mathrm{SU}}_n}$ is the Haar probability measure. Our second main result is about the random walks generated by ${\tau }_{w,{\mathrm{SU}}_n}$. We provide exponential upper bounds on the large Fourier coefficients of ${\tau }_{w,{\mathrm{SU}}_n}$, and as a consequence we show there exists $t(w)\in \mathbb{N}$, such that ${\tau }_{w,{\mathrm{SU}}_n}^{\ast t}$ has bounded density for every $t\ge t(w)$ and every $n\ge 2$, answering a conjecture by the first two authors. As a key step in the proof, we establish, for every large irreducible character $\rho$ of ${\mathrm{SU}}_n$, an exponential upper bound of the form $\left|\rho (g)\right|<\rho (1{)}^{1-\epsilon }$, for elements $g$ in ${\mathrm{SU}}_n$ whose eigenvalues are sufficiently spread out on the unit circle in ${\mathbb{C}}^{\times }$.

36 pages