We study the distribution of signs of the (real-valued) Hecke eigenvalues $A(m,1,...,1)$ of self-dual Hecke--Maass cusp forms for the group $\mathrm{SL}_n(\mathbb Z)$, where $n\geq 2$ is an integer. Our main result establishes, under Generalised Ramanujan--Petersson Conjecture, that for almost all $x\sim X$ the short interval $[x,x+H]$ contains a subset $\mathcal S$ (resp. $\mathcal S')$ of size $\gg H(\log X)^{1/n^2-1}$ such that $A(m,1,...,1)$ is positive (resp. negative) for all $m\in\mathcal S$ (resp. $m\in\mathcal S'$), provided that $(\log X)^{1-1/n^2}\ll H\ll X$. We also prove a slightly stronger result unconditionally for $\mathrm{GL}_2$ and $\mathrm{GL}_3$ Hecke--Maass cusp forms. In addition, we obtain results under weaker bounds towards Generalised Ramanujan--Petersson Conjecture. Finally, as a by-product of our methods, we improve earlier bounds for the number of Hecke eigenvalues of same sign also in long intervals unconditionally for $\mathrm{GL}_2$ and $\mathrm{GL}_3$ Hecke--Maass cusp forms, and under Generalised Ramanujan--Petersson Conjecture for $\mathrm{GL}_n$ forms when $n\geq 4$.
Let $(\mathfrak g, \mathfrak k)$ be the complexication of a Hermitian symmetric pair and let $\gamma_1<\cdots<\gamma_r$ be the Harish-Chandra strongly orthogonal roots, with $\gamma_1$ the lowest non-compact root, and let $\zeta$ be the normalized central character of $\mathfrak k$. We consider the unitary highest weig...
We show that for a fixed $\mathrm{SL}_3(\mathbb Z)$ Hecke--Maass cusp form $F$, there are infinitely many primitive Dirichlet characters $\chi$ such that $L(1/2, F\times \chi)$ and $L(1/2, \chi)$ do not vanish \emph{simultaneously}. More precisely, we obtain an asympotic formula with a {\em power-saving} error term for...
Let $H_n(q)$ be the generic Iwahori--Hecke algebra associated to the symmetric group $\mathfrak{S}_n$ of degree $n$ and let $\mathscr{Y}_{n,i}$ be the sum of standard basis of $H_n(q)$ with Coxeter length $i$ for $1\leq i\leq \binom{n}{2}$. We show that $\mathscr{Y}_{n,1}$ and $\mathscr{Y}_{n,2}$ are noncommutative whe...
In this article, we first prove that the Hecke operator $T_t$ has no eigenform in $\Sla$ with eigenvalue $-t^{k/2}$, when the characteristic of the base field is odd. Furthermore, if $\dim \Slt$ is even, we show that $T_t$ has no eigenform in $\Sla$ with eigenvalue $t^{k/2}$. As a consequence, we prove that the direct...
Let $S_k$ be the space of cusp forms of weight $k$ for ${\rm SL}_2(\mathbb Z)$, with $\dim S_k\geq1$, and let $\chi_p=\left(\frac{\cdot}{p}\right)$ for odd primes $p$. For fixed $s$ with ${\rm Re}(s)>(k+1)/2$, we prove that the proportion of primes $p\leq X$ satisfying $L(h,\chi_p,s)=0$ tends to zero as $X\to\infty$, u...
Let $M=\widetilde H^*(MO(2);\mathbb F_2)$ and let $\mathcal A(1)$ be the subalgebra of the mod $2$ Steenrod algebra generated by $\operatorname{Sq}^1$ and $\operatorname{Sq}^2$. In this work, we determine the quotient $\mathbb F_2\otimes_{\mathcal A(1)}M$ in every internal degree $h\geq0$, where internal degree $h$ cor...
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