arXiv subjects

Jakub Löwit

Publications and source records attributed to Jakub Löwit.

4 recordsLinked to original sources

arXiv2026

Equivariant $K$-theory ring of the affine Grassmannian as deformation to normal cone

We revisit fixed-point localization techniques in equivariant topological $K$-theory, presenting its fibers over varying points of the equivariant base as cohomology of the corresponding fixed-point schemes. We then employ this framework for an effective description of the equivariant $K$-theory ring of the affine Grassmannian $\mathcal{G}r_G$. We do so by a detailed study of the relevant fixed points, extending known results to the action of any $(x, ζ) \in T \times \mathbb{G}_{m}^{\mathrm{rot}}$. We obtain a description of the complexified topological $K$-theory ring $K^{\mathrm{top}, 0}_{T\times \mathbb{G}_{m}^{\mathrm{rot}}}(\mathcal{G}r_G; \mathbb{C})$ as the ring of functions on a family of affine blowups, extending and proving a conjecture of Roman Bezrukavnikov. While this ring is fairly big, we provide a preferred infinite set of topological generators.

math.AG↗

arXiv2026

Equivariant localizing invariants of simple varieties

We define a certain class of simple varieties over a field $k$ by a constructive recipe and show how to control their (equivariant) truncating invariants. Consequently, we prove that on simple varieties: (i) if $k=\overline{k}$ and $\mathrm{char} \ k = p$, the $p$-adic cyclotomic trace is an equivalence; (ii) if $k = \mathbb{Q}$, the Goodwillie-Jones trace is an isomorphism in degree zero; (iii) we can control homotopy invariant $K$-theory $KH$, which is equivariantly formal and determined by its topological counterparts. Simple varieties are quite special, but encompass important singular examples appearing in geometric representation theory. We in particular show that both finite and affine Schubert varieties for $GL_n$ lie in this class, so all the above results hold for them.

math.AG↗

arXiv2026

Equivariant $K$-theory, affine Grassmannian and perfection

We study torus-equivariant algebraic $K$-theory of affine Schubert varieties in the perfect affine Grassmannians over $\mathbb{F}_p$. We further compare it to the torus-equivariant Hochschild homology of perfect complexes, which has a geometric description in terms of global functions on certain fixed-point schemes. We prove that $\mathbb{F}_p$-linearly, this comparison is an isomorphism. Our approach is quite constructive, resulting in new computations of these $K$-theory rings. We establish various structural results for equivariant perfect algebraic $K$-theory on the way; we believe these are of independent interest.

math.AG↗

arXiv2024

On modulo $\ell$ cohomology of $p$-adic Deligne-Lusztig varieties for $GL_n$

In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a $p$-adic version of this theory started to emerge: there are $p$-adic Deligne-Lusztig spaces, whose cohomology encodes representation theoretic information for $p$-adic groups - for instance, it partially realizes local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic $\ell \neq p$ has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain $p$-adic Deligne-Lusztig spaces to Vignéras's modular local Langlands correspondence for $\mathbf{GL}_n$.

math.RT↗