Evaluating Characteristic Functions of Character Sheaves at Unipotent Elements

Assume $\mathbf{G}$ is a connected reductive algebraic group defined over an algebraic closure $\mathbb{K} = \overline{\mathbb{F}}_p$ of the finite field of prime order $p>0$. Furthermore, assume that $F : \mathbf{G} \to \mathbf{G}$ is a Frobenius endomorphism of $\mathbf{G}$. In this article we give a formula for the value of any $F$-stable character sheaf of $\mathbf{G}$ at a unipotent element. This formula is expressed in terms of class functions of $\mathbf{G}^F$ which are supported on a single unipotent class of $\mathbf{G}$. In general these functions are not determined, however we give an expression for these functions under the assumption that $Z(\mathbf{G})$ is connected, $\mathbf{G}/Z(\mathbf{G})$ is simple and $p$ is a good prime for $\mathbf{G}$. In this case our formula is completely explicit.Comment: 29 pages. Parts of this article first appeared in arXiv:1306.5882. This is an expanded and generalised of version of what appears there. (v2): 30 pages. Final version post referees report. Referenced work of Digne-Lehrer-Michel who also independently obtained Theorem 7.

A Note on Skew Characters of Symmetric Groups

In previous work Regev used part of the representation theory of Lie superalgebras to compute the values of a character of the symmetric group whose decomposition into irreducible constituents is described by semistandard $(k,\ell)$-tableaux. In this short note we give a new proof of Regev's result using skew characters.Comment: 4 page

Induced Characters of Type D Weyl Groups and the Littlewood-Richardson Rule

For any ordinary irreducible character of a maximal reflection subgroup of type $D_aD_b$ of a type $D$ Weyl group we give an explicit decomposition of the induced character in terms of Littlewood-Richardson coefficients.Comment: 8 pages; (v2) made some typographical changes and added the type D branching rule as a corollary; (v3) title changed and various other changes made based on referees report. We thank the referee for their careful reading of the manuscript and useful commments; (v4) fixed an annoying bug in the reference

On The Mackey Formula for Connected Centre Groups

Let $\mathbf{G}$ be a connected reductive algebraic group over $\overline{\mathbb{F}}_p$ and let $F : \mathbf{G} \to \mathbf{G}$ be a Frobenius endomorphism endowing $\mathbf{G}$ with an $\mathbb{F}_q$-rational structure. Bonnaf\'e--Michel have shown that the Mackey formula for Deligne--Lusztig induction and restriction holds for the pair $(\mathbf{G},F)$ except in the case where $q = 2$ and $\mathbf{G}$ has a quasi-simple component of type $\sf{E}_6$, $\sf{E}_7$, or $\sf{E}_8$. Using their techniques we show that if $q = 2$ and $Z(\mathbf{G})$ is connected then the Mackey formula holds unless $\mathbf{G}$ has a quasi-simple component of type $\sf{E}_8$. This establishes the Mackey formula, for instance, in the case where $(\mathbf{G},F)$ is of type $\sf{E}_7(2)$. Using this, together with work of Bonnaf\'e--Michel, we can conclude that the Mackey formula holds on the space of unipotently supported class functions if $Z(\mathbf{G})$ is connected.Comment: 7 pages; v2., minor changes, added Lemma 3.4 for clarit

Lusztig Induction, Unipotent Supports, and Character Bounds

Recently, a strong exponential character bound has been established in [3] for all elements $g \in \mathbf{G}^F$ of a finite reductive group $\mathbf{G}^F$ which satisfy the condition that the centraliser $C_{\mathbf{G}}(g)$ is contained in a $(\mathbf{G},F)$-split Levi subgroup $\mathbf{M}$ of $\mathbf{G}$ and that $\mathbf{G}$ is defined over a field of good characteristic. In this paper, assuming a weak version of Lusztig's conjecture relating irreducible characters and characteristic functions of character sheaves holds, we considerably generalize this result by removing the condition that $\mathbf{M}$ is split. This assumption is known to hold whenever $Z(\mathbf{G})$ is connected or when $\mathbf{G}$ is a special linear or symplectic group and $\mathbf{G}$ is defined over a sufficiently large finite field.Comment: 35 pages; v2. minor improvements to abstract and introduction; v3. further improvements to the exposition; v4. significant changes. Main result now works for special linear and symplectic groups. Added results on groups of type A generalising results of Hildebrand; v5. post referee repor

Principal 22-Blocks and Sylow 22-Subgroups

Let $G$ be a finite group with Sylow $2$-subgroup $P \leqslant G$. Navarro-Tiep-Vallejo have conjectured that the principal $2$-block of $N_G(P)$ contains exactly one irreducible Brauer character if and only if all odd-degree ordinary irreducible characters in the principal $2$-block of $G$ are fixed by a certain Galois automorphism $\sigma \in \mathrm{Gal}(\mathbb{Q}_{|G|}/\mathbb{Q})$. Recent work of Navarro-Vallejo has reduced this conjecture to a problem about finite simple groups. We show that their conjecture holds for all finite simple groups, thus establishing the conjecture for all finite groups.Comment: 12 page

The Trilemma in History: Tradeoffs among Exchange Rates, Monetary Policies, and Capital Mobility

The exchange-rate regime is often seen as constrained by the monetary policy trilemma, which imposes a stark tradeoff among exchange stability, monetary independence, and capital market openness. Yet the trilemma has not gone without challenge. Some (e.g., Calvo and Reinhart 2001, 2002) argue that under the modern float there could be limited monetary autonomy. Others (e.g., Bordo and Flandreau 2003), that even under the classical gold standard domestic monetary autonomy was considerable. This paper studies the coherence of international interest rates over more than 130 years. The constraints implied by the trilemma are largely borne out by history.

Monetary Sovereignty, Exchange Rates, and Capital Controls: The Trilemma in the Interwar Period

The interwar period was marked by the end of the classical gold standard regime and new levels of macroeconomic disorder in the world economy. The interwar disorder often is linked to policies inconsistent with the constraint of the open-economy trilemmathe inability of policymakers simultaneously to pursue a fixed exchange rate, open capital markets, and autonomous monetary policy. The first two objectives were linchpins of the pre-1914 order. As increasingly democratic polities faced pressures to engage in domestic macroeconomic management, however, either currency pegs or freedom of capital movements had to yield. This historical analytic narrative is compellingwith significant ramifications for today's world, if truebut empirically controversial. We apply theory and empirics to the interwar data and find strong support for the logic of the trilemma. Thus, an inability to pursue consistent policies in a rapidly changing political and economic environment appears central to an understanding of the interwar crises, and the same constraints still apply today.