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1. Non-vanishing of geometric Whittaker coefficients for reductive groups

   https://doi.org/10.1090/jams/1051  

   Færgeman, Joakim; Raskin, Sam (April 2025, Journal of the American Mathematical Society)

   We prove that cuspidal automorphic $D$-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from $G{L}_n$to general reductive groups. The key tool is a microlocal interpretation of Whittaker coefficients. We establish various exactness properties in the geometric Langlands context that may be of independent interest. Specifically, we show Hecke functors are $t$-exact on the category of tempered $D$-modules, strengthening a classical result of Gaitsgory (with different hypotheses) for $G{L}_n$. We also show that Whittaker coefficient functors are $t$-exact for sheaves with nilpotent singular support. An additional consequence of our results is that the tempered, restricted geometric Langlands conjecture must be $t$-exact. We apply our results to show that for suitably irreducible local systems, Whittaker-normalized Hecke eigensheaves are perverse sheaves that are irreducible on each connected component of ${\mathrm{Bun}}_G$. 

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2. Equivariant prime ideals for infinite dimensional supergroups

   https://doi.org/10.1090/tran/9202  

   Laudone, Robert; Snowden, Andrew (September 2024, Transactions of the American Mathematical Society)

   Let $A$be a commutative algebra equipped with an action of a group $G$. The so-called $G$-primes of $A$are the equivariant analogs of prime ideals, and of central importance in equivariant commutative algebra. When $G$is an infinite dimensional group, these ideals can be very subtle: for instance, distinct $G$-primes can have the same radical. In previous work, the second author showed that if $G={\mathbf{G}\mathbf{L}}_{\mathrm{\infty }}$and $A$is a polynomial representation, then these pathologies disappear when $G$is replaced with the supergroup ${\mathbf{G}\mathbf{L}}_{\mathrm{\infty }|\mathrm{\infty }}$and $A$with a corresponding algebra; this leads to a geometric description of $G$-primes of $A$. In the present paper, we construct an abstract framework around this result, and apply the framework to prove analogous results for other (super)groups. We give some applications to the isomeric determinantal ideals (commonly known as “queer determinantal ideals”). 

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3. A Volume = Multiplicity formula for p-families of ideals

   https://doi.org/10.1090/proc/16451  

   Das, Sudipta (October 2023, Proceedings of the American Mathematical Society)

   In this article, we work with certain families of ideals called $p$-families in rings of prime characteristic. This family of ideals is present in the theories of tight closure, Hilbert-Kunz multiplicity, and $F$-signature. For each $p$-family of ideals, we attach an Euclidean object called $p$-body, which is analogous to the Newton Okounkov body associated with a graded family of ideals. Using the combinatorial properties of $p$-bodies and algebraic properties of the Hilbert-Kunz multiplicity, we establish a Volume = Multiplicity formula for $p$-families of ${\mathfrak{m}}_R$-primary ideals in a Noetherian local ring $R$. 

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4. Compatible ideals in ℚ-Gorenstein rings

   https://doi.org/10.1090/proc/16331  

   Polstra, Thomas; Schwede, Karl (October 2023, Proceedings of the American Mathematical Society)

   Suppose $R$is a $F$-finite and $F$-pure $\mathbb{Q}$-Gorenstein local ring of prime characteristic $p>0$. We show that an ideal $I\subseteq <\#comment/>R$is uniformly compatible ideal (with all $p^{-<\#comment/>e}$-linear maps) if and only if exists a module finite ring map $R\to <\#comment/>S$such that the ideal $I$is the sum of images of all $R$-linear maps $S\to <\#comment/>R$. In other words, the set of uniformly compatible ideals is exactly the set of trace ideals of finite ring maps. 

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5. Residual intersections and linear powers

   https://doi.org/10.1090/btran/127  

   Eisenbud, David; Huneke, Craig; Ulrich, Bernd (October 2023, Transactions of the American Mathematical Society, Series B)

   If $I$is an ideal in a Gorenstein ring $S$, and $S/I$is Cohen-Macaulay, then the same is true for any linked ideal $I^{\prime }$; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal $L_n$of minors of a generic $2\times <\#comment/>n$matrix when $n>3$. In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of $I$. For example, suppose that $K$is the residual intersection of $L_n$by $2n-<\#comment/>4$general quadratic forms in $L_n$. In this situation we analyze $S/K$and show that $I^{n-<\#comment/>3}\left(S/K\right)$is a self-dual maximal Cohen-Macaulay $S/K$-module with linear free resolution over $S$. The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented. 

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