Free monodromic Hecke categories and their categorical traces

The goal of this paper is to give a new construction of the free monodromic categories defined by Yun. We then use this formalism to give simpler constructions of the free monodromic Hecke categories and then compute the trace of Frobenius and of the identity on them. As a first application of the formalism, we produce new proofs of key theorems in Deligne–Lusztig theory.

💡 Research Summary

The paper presents a new construction of the free monodromic categories originally introduced by Yun, and uses this framework to give streamlined definitions of free monodromic Hecke categories and to compute their categorical traces. The authors work over an algebraically closed field of characteristic p>0, fix a torus T, and introduce the condensed ring R_T,Λ built from the maximal pro‑ℓ quotient of the étale fundamental group of T. This ring serves as the coefficient ring for a rank‑one locally constant sheaf L_T on T, called the free monodromic unipotent local system. By considering (T, L_T)-equivariant sheaves on a scheme X, they define the category D_ic(X,R_T)_{unip}. Theorem 1.1.3 (Theorem 2.7.1) establishes a natural equivalence between this category and Yun’s original free monodromic unipotent category, valid for all coefficient rings Λ∈{ℤ_ℓ, 𝔽_ℓ, ℚ_ℓ} and even for twisted sheaves via a suitable completion.

With this formalism in place, the authors construct the free monodromic Hecke category H associated to a (possibly disconnected) reductive group G with a Borel pair B=TU. The stack U\G/U carries three torus actions (left T, right T, and simultaneous T×T). By taking (T×T)-equivariant free monodromic sheaves, they obtain a monoidal category whose product is defined by a convolution diagram involving the multiplication map m: U\G×U G/U → U\G/U. Crucially, the new construction restores full left–right symmetry, making the monoidal structure immediate. The authors prove that H is compactly generated, quasi‑rigid, a direct sum of rigid categories, and possesses a canonical pivotal structure (Lemmas 4.1.2, Theorem 4.4.2, Corollary 4.4.9), mirroring results of Bezrukavnikov–Yun–Zhang but in the free monodromic setting.

The paper then turns to categorical traces. For the identity endofunctor, the trace Tr(id, H) coincides with the categorical center Z(H), which the authors identify with the category of free‑monodromic character sheaves CS^∧. They show that CS^∧ is equivalent to modules over a certain algebra A inside D(G Ad G, R_T×T), thereby recovering Lusztig’s classical character sheaf theory when G is connected with connected center. For the Frobenius endomorphism F, Theorem 6.1.1 proves a canonical isomorphism \