arXiv subjects
Publications and source records attributed to Alexander Hazeltine.
For symplectic and even orthogonal groups over a $p$-adic field, we determine the number of local Arthur packets containing the local theta lift of a tempered representation at the first occurrence in the going-up tower. These counts show that the local theta lifts may lie in many more local Arthur packets than those predicted by the Adams conjecture.
In [HJLLZ24], we proposed a new conjecture on the structure of the unitary dual of connected reductive groups over non-Archimedean local fields of characteristic zero based on their Arthur representations and verified it for all the known cases on the unitary dual problem. One step towards this conjecture involves the question whether certain complementary Arthur representations are unitary. In this paper, we give an explicit characterization of the complementary Arthur representations for symplectic and split odd special orthogonal groups. As applications, we obtain interesting constraints on local components of irreducible self-dual cuspidal automorphic representations of $\mathrm{GL}_N$, especially when $N=2,3$.
In this paper we study the upper bound of wavefront sets of irreducible admissible representations of connected reductive groups defined over non-Archimedean local fields of characteristic zero. We formulate a new conjecture on the upper bound and show that it can be reduced to that of anti-discrete series representations, namely, those whose Aubert-Zelevinsky duals are discrete series. Then, we show that this conjecture is equivalent to the Jiang conjecture on the upper bound of wavefront sets of representations in local Arthur packets and also equivalent to an analogous conjecture on the upper bound of wavefront sets of representations in local ABV packets.
We give an algorithm to compute the Pyasetskii involution for $\mathrm{Sp}_{2n}$, $\mathrm{SO}_{2n+1}$ and $\mathrm{O}_{2n}$. The algorithm is a combination of Moeglin-Waldspurger's algorithm for the Pyasetskii involution for $\mathrm{GL}_n$ ([MW86]) and Lanard-M${í}$nguez's algorithm for the Aubert-Zelevinsky involution of bad parity representations for classical groups ([LM25]). In particular, we give a geometric interpretation of the bad parity case of Lanard-M${í}$nguez's algorithm.
We study the functoriality of the local theta correspondence for classical $p$-adic groups. This is realized via the adaptation of the Adams conjecture to ABV-packets. We provide evidence for the conjecture, especially in the case of general linear groups.
We determine the number of local Arthur packets containing a certain fixed tempered representation for classical $p$-adic groups. More specifically, given a tempered extended multi-segment supported in the integers, we determine a count for all extended multi-segments which arise from it through applications of the operators arising from the theory of intersections of local Arthur packets.
In this paper, we propose a new conjecture describing the structure of the unitary dual in terms of Arthur representations for connected reductive algebraic groups defined over any non-Archimedean local field of characteristic zero. This conjecture provides a candidate set for the unitary dual, constructed from Arthur representations. For classical groups, we develop an explicit algorithm to generate this candidate set. Evidence for its exhaustiveness includes compatibility with the known generic unitary dual, unramified unitary dual, and low-corank representations. As further support, we verify the conjecture for the unitary dual of the exceptional group of type $G_2$.
We prove a converse theorem for the case of quasi-split non-split even special orthogonal groups over finite fields. There are two main difficulties which arise from the outer automorphism and non-split part of the torus. The outer automorphism is handled similarly to the split case, while new ideas are developed to overcome the non-split part of the torus.
We give a direct proof of the local converse theorem for quasi-split non-split $\mathrm{SO}_{2l}$ over a local non-Archimedean field of characteristic $p\neq 2$, applying the theory of Howe vectors and partial Bessel functions.
In this paper, applying the intersection theory of local Arthur packets, for symplectic and split odd special orthogonal groups G_n, we give the first complete proof of the enhanced Shahidi conjecture on generic representations in local Arthur packets. We also classify unramified representations of Arthur type for G_n, and show that they lie in exactly one local Arthur packet, which is anti-generic. Then, we discuss the global applications of these results.
In this paper, for symplectic and split odd special orthogonal groups, we develop an account of theory on the intersection problem of local Arthur packets. Specifically, following Atobe's reformulation on Mœglin's construction of local Arthur packets, we give a complete set of operators on the construction data, based on which, we provide algorithms and Sage codes to determine whether a given representation is of Arthur type. Furthermore, for any representation $π$ of Arthur type, we give a precise formula for the set $$ Ψ(π)=\{ \text{local Arthur parameter }ψ\ | \ \text{the local Arthur packet } Π_ψ \text{ contains } π\}.$$ Our results have many applications, including the precise counting of tempered representations in any local Arthur packet, specifying and characterizing "the" local Arthur parameter in $Ψ(π)$ for $π$, especially when $π$ belongs to several local Arthur packets but does not belong to any local $L$-packet of Arthur type.
In this paper, we prove the closure ordering conjecture on the local $L$-parameters of representations in local Arthur packets of $\mathrm{G}_n=\mathrm{Sp}_{2n}, \mathrm{SO}_{2n+1}$ over a non-Archimedean local field of characteristic zero. Precisely, given any representation $π$ in a local Arthur packet $Π_ψ$, the closure of the local $L$-parameter of $π$ in the Vogan variety must contain the local $L$-parameter corresponding to $ψ$. This conjecture reveals a geometric nature of local Arthur packets and is inspired by the work of Adams, Barbasch, and Vogan, and the work of Cunningham, Fiori, Moussaoui, Mracek, and Xu, on ABV-packets. As an application, for general quasi-split connected reductive groups, we show that the closure ordering conjecture implies the enhanced Shahidi conjecture, under certain reasonable assumptions. This provides a framework towards the enhanced Shahidi conjecture in general. We verify these assumptions for $\mathrm{G}_n$, hence give a new proof of the enhanced Shahidi conjecture. At last, we show that local Arthur packets cannot be fully contained in other ones, which is in contrast to the situation over Archimedean local fields and has its own interests.
The Adams conjecture states that the local theta correspondence sends a local Arthur packet to another local Arthur packet. Mœglin confirmed the conjecture when lifting to groups of sufficiently high rank and also showed that it fails in low rank. Recently, Bakić and Hanzer described when the Adams conjecture holds in low rank for a representation in a fixed local Arthur packet. However, a representation may lie in many local Arthur packets and each gives a minimal rank for which the Adams conjecture holds. In this paper, we study the interplay of intersections of local Arthur packets with the Adams conjecture.
In this paper, we prove the local converse theorem for split even special orthogonal groups over a non-Archimedean local field of characteristic zero. This is the only case left on local converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. We apply new ideas of considering the summation of partial Bessel functions and overcome this difficulty. As a direct application, we obtain a weak rigidity theorem for irreducible generic cuspidal representations of split even special orthogonal groups.
We prove a converse theorem for split even special orthogonal groups over finite fields. This is the only case left on converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. In this paper, we develop new ideas and overcome this difficulty.