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September 10, 2025

On the unramified Eisenstein spectrum

Key Points

The analysis determines the spectrum of the Hecke algebra from unramified Eisenstein series in the context of minimal parabolic groups.
Key findings show that the spectral decomposition relates closely to the structure of the Springer stack and its implications.
Utilizing a decomposition of the Springer stack, the research explores the Langlands dual group through derived quotient stacks.
This research highlights the interaction between Hecke algebras and derived categories in algebraic geometry, expanding theoretical frameworks.

Abstract

For a split reductive group L G ^ {L} G over a global field, we determine the spectrum of the spherical Hecke algebra coming from the unramified Eisenstein series for the minimal parabolic L B ^ {L} B. This is done using a certain decomposition of the Springer stack T ∗ (B ∖ G / B) T^* (B G/ B) for the Langlands dual group G G in the additive cobordism group of cohomologically proper derived quotient stacks.

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