Research

Papers

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

   Let \(G=\textbf G(F)\) 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 \(q\to\infty\), 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.

8. 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 \(\widetilde{\mathcal N}\to\mathcal N\) be the Springer resolution and let \(\widetilde U\to U\) be the restriction to \(U\). We characterize the irreducible objects in the exotic \(t\)-structure of \(D^{\mathrm b}\mathrm{Coh}^{G^\vee}(\widetilde U)\), 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 \(\mathcal O_{\widetilde U}(\lambda)\) in the equivariant \(K\)-theory \(K^{G^\vee}(\widetilde U)\) in terms of the irreducible objects, which computes Kazhdan-Lusztig polynomials.

9. 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 \(\varphi_\sigma\) 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 \(\varphi_\sigma\). 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 \(\varphi_\sigma\) in all places not over \(p\), and under some additional hypotheses we can completely determine their relationship.

10. 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.

11. 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 \(\mathrm{Sp}_4\).

12. 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.

13. 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