Research

Papers

8. Geometric characterization of the group law in the Weyl group [arXiv:2503.19645]

   For a reductive group $G$, it has a Weyl group $W$. Usually it is defined as $N_G(T)/T$ for a maximal torus $T$, and any two maximal tori $T_1$ and $T_2$ are conjugate, so there is an isomorphism $N_G(T_1)/T_1\simeq N_G(T_2)/T_2$, but this isomorphism is only unique up to conjugation by $W$. A more canonical definition of $W$ is as the set of $G$-orbits in the variety $\mathcal{B}\times \mathcal{B}$, where $\mathcal{B}$ is the flag variety of $G$. A natural question now is: how to characterize the multiplication law in $W$? We prove that given two $G$-orbits $O(w_1)$ and $O(w_2)$ in $\mathcal{B}\times \mathcal{B}$, the convolution $O(w_1)\ast O(w_2)$ has a unique closed $G$-orbit $O(w_1{w}_2)$.

9. Trace Paley-Wiener theorem for Braverman-Kazhdan's asymptotic Hecke algebra [arXiv:2407.02752]

   Let $G=\mathbf{\text{G}}(F)$ be a $p$-adic reductive group, with Hecke algebra $\mathcal{H}(G)$. Then the structure of the Iwahori Hecke algebra $\mathcal{H}(G,I)$ only depends on the size $q$ of the residue field of $F$. Lusztig defines the asymptotic Hecke $J$, which is a "limit" of $\mathcal{H}(G,I)$ as $q\to \mathrm{\infty }$, and can be used to understand the Iwahori Hecke algebra. Although the full Hecke algebra $\mathcal{H}(G)$ no longer just depends on the parameter $q$, Braverman and Kazhdan have extended the definition of $J$ to the full asymptotic Hecke algebra $\mathcal{J}(G)$. One nice feature of Lusztig's $J$ is that its irreducible representations correspond to standard representations of the Iwahori Hecke algebra. We prove an analogous statement for $\mathcal{J}(G)$. In doing so, we confirm a conjecture of Bezrukavnikov, Braverman, and Kazhdan that $\mathcal{H}(G)\to \mathcal{J}(G)$ induces an isomorphism on the cocenters.

10. Affine Kazhdan-Lusztig polynomials on the subregular cell in non simply-laced Lie algebras: with an application to character formulae, with Vasily Krylov (with an appendix by Roman Bezrukavnikov, Vasily, and myself.) [arXiv:2401.06605]

    Bezrukavnikov, Kac, and Krylov uses the geometry of the Springer fiber ${\mathcal{B}}_e$ where $e$ is a subregular nilpotent element to compute special values of affine Kazhdan-Lusztig polynomials when $G$ is simply-laced. We extend the computation to non-simply-laced Lie groups $G$. In particular, letting $U\subset \mathcal{N}$ be the subset of the nilpotent cone of ${\mathfrak{g}}^{\vee }$ consisting of regular and subregular nilpotent orbits. Let $\tilde{\mathcal{N}}\to \mathcal{N}$ be the Springer resolution and let $\tilde{U}\to U$ be the restriction to $U$. We characterize the irreducible objects in the exotic $t$-structure of $D^{\mathrm{b}}{\mathrm{Coh}}^{G^{\vee }}(\tilde{U})$, which is the categorification of the canonical basis of the anti-spherical representation of ${\mathcal{H}}_{\ge \mathrm{subreg}}$. We then compute the class of the standard basis ${\mathcal{O}}_{\tilde{U}}(\lambda )$ in the equivariant $K$-theory $K^{G^{\vee }}(\tilde{U})$ in terms of the irreducible objects, which computes Kazhdan-Lusztig polynomials.

11. Rationality of the Local Jacquet-Langlands Correspondence for $\mathrm{GL}(n)$ [arXiv:2307.06039]

    Let $F/{\mathbb{Q}}_p$ be a finite extension and let $D/F$ be a central division algebra. Then to a representation $\sigma$ of $D^{\times }$ we may attach an $L$-parameter, a $n$-dimensional representation ${\phi }_{\sigma }$ of the Weil group $W_F$. Now the field of rationality is the field generated by all character values of these representations, and the field of rationality of $\sigma$ equals the field of rationality of ${\phi }_{\sigma }$. However, representations need not be defined over their field of rationality: for example, the irreducible two-dimensional representation of $S_3$ has field of rationality $\mathbb{Q}$ but it cannot be defined over $\mathbb{Q}$. The field of definition is controlled by the Hasse invariant. We determine the relationship between the Hasse invariant of $\sigma$ and ${\phi }_{\sigma }$ in all places not over $p$, and under some additional hypotheses we can completely determine their relationship.

12. The explicit local Langlands correspondence for $G_2$ II, with Yujie Xu [arXiv:2304.02630]

    We write down some character formulas for representations of $G_2(F)$ considered by Aubert-Xu, and shows that stability for $L$-packets uniquely pins down their Local Langlands correspondence.

13. Explicit local Langlands correspondence for ${\mathrm{GSp}}_4$ and ${\mathrm{Sp}}_4$, with Yujie Xu [arXiv:2304.02622]

    We give a purely local, explicit construction of the Local Langlands correspodnence for ${\mathrm{GSp}}_4$ and ${\mathrm{Sp}}_4$.

14. Gelfand-Kirillov dimension of representations of ${\mathrm{GL}}_n$ over a nonarchimedean local field, as part of the SPUR program, mentored by Hao Peng [arXiv:2208.05139]

    Let $\pi$ be an irreducible representation of ${\mathrm{GL}}_n(F)$ where $F$ is a non-archimedean local field, and let $K_N=1+M_n({\mathfrak{p}}^N)$, which are a decreasing sequence of compact open subgroups of ${\mathrm{GL}}_n(F)$. Then the space ${\pi }^{K_N}$ of $K_N$-fixed vectors is a finite-dimensional vector space, and it grows as a polynomial of $q^N$ for large $N$. The Gelfand-Kirillov dimension of $\pi$ is the degree of this polynomial. We explicitly compute the Gelfand-Kirillov dimension for arbitrary irreducible representations of ${\mathrm{GL}}_n(F)$ using Zelevinsky's classification.

15. Mermorphic functions with the same preimages at several finite sets, as part of the PRIMES program, mentored by Professor Michael Zieve [link]

    Nevanlinna showed that a nonconstant meromorphic function on $\mathbb{C}$ is uniquely determined by its preimage at any five points: i.e., that for functions $p$ and $q$ and points $z_1,\dots ,z_5\in \mathbb{C}$, if $p^{-1}(z_i)=q^{-1}(z_i)$ then $p=q$. We generalize this to pre-images of sets, and show that if $p^{-1}(S_i)=q^{-1}(S_i)$ counting multiplicities, for subsets $S_1,\dots ,S_4$ where no subset is contained in the union of the other three, then there exists a non-constant rational function $g$ such that $g\circ p=g\circ q$, with some explicit bounds on the degree of $g$. We also prove analogous statements for equations of the form $p^{-1}(S_i)=q^{-1}(T_i)$, where now $S_i$ and $T_i$ need not be equal.

Talks

4. Academia Sinica Seminar in Representation Theory, Cocenter of Braverman-Kazhdan's Asymptotic Hecke Algebra, December 2024
5. Yale Geometry, Symmetry, and Physics Seminar, Trace Paley-Wiener theorem for Braverman-Kazhdan's Asymptotic Hecke Algebra, September 2024
6. MIT Infinite-dimensional Algebra Seminar, Affine Kazhdan-Lusztig polynomials on the subregular cell: with an application to character formulae, February 2024
7. MIT Lie Groups Seminar, The explicit Local Langlands Correspondence for $G_2$ and ${\mathrm{GSp}}_4$, character formulas and stability, October 2023