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March 2, 2026Open Access

Unified Modular Singular Support for the Adjoint Quotient Stack, with Extensions to Bad Primes, Quasi-Smooth Derived Stacks, and Explicit Motivic Realizations for Tame (GLₙ)

Key Points

The aim is to establish a comprehensive modular singular-support theory for the adjoint quotient stack and explore its extensions.
Developed two independent constructions for modular singular-support theory.
Identified nilpotent spectral categories through Springer generation.
Established derived Morita equivalence and formality in the modular spectral context.
Provided a tame depth-zero realization for (GL_n) over finite fields.
Unified modular singular support was successfully proven for the adjoint quotient stack.
Extensions to bad primes and selected quasi-smooth derived stacks were established.
Explicit results include motivic Euler-products and fiber-factorizations.
Identified exotic nilpotent cone properties related to type (C) at ℓ=2.

Abstract

We prove a unified modular singular-support theory for the adjoint quotient stack by two independent constructions, identify the nilpotent spectral category by Springer generation, establish formality and derived Morita equivalence on the modular spectral side, and give an explicit tame depth-zero (GLₙ) finite-field realization together with motivic Euler-product and fiber-factorization results. The paper also includes extensions to bad primes, selected quasi-smooth derived stacks, and the exotic nilpotent cone in type (C) at (=2).

Unified Modular Singular Support for the Adjoint Quotient Stack, with Extensions to Bad Primes, Quasi-Smooth Derived Stacks, and Explicit Motivic Realizations for Tame (GLₙ)

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Abstract

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