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1. Cohomological representations of parahoric subgroups

   https://doi.org/10.1090/ert/557  

   Chan, Charlotte; Ivanov, Alexander (January 2021, Representation Theory of the American Mathematical Society)

   We give a geometric construction of representations of parahoric subgroups P P of a reductive group G G over a local field which splits over an unramified extension. These representations correspond to characters θ \theta of unramified maximal tori and, when the torus is elliptic, are expected to give rise to supercuspidal representations of G G . We calculate the character of these P P -representations on a special class of regular semisimple elements of G G . Under a certain regularity condition on θ \theta , we prove that the associated P P -representations are irreducible. This generalizes a construction of Lusztig from the hyperspecial case to the setting of an arbitrary parahoric. 

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2. Regular Supercuspidal Representations

   Tasho Kaletha (July 2019, Journal of the American Mathematical Society)

   We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups. 

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3. ON THE EXISTENCE OF ADMISSIBLE SUPERSINGULAR REPRESENTATIONS OF -ADIC REDUCTIVE GROUPS

   https://doi.org/10.1017/fms.2019.50  

   HERZIG, FLORIAN; KOZIOŁ, KAROL; VIGNÉRAS, MARIE-FRANCE (January 2020, Forum of Mathematics, Sigma)

   Suppose that $$\mathbf{G}$$ is a connected reductive group over a finite extension $$F/\mathbb{Q}_{p}$$ and that $$C$$ is a field of characteristic  $$p$$ . We prove that the group $$\mathbf{G}(F)$$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over  $$C$$ . 

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4. Automorphy lifting with adequate image

   https://doi.org/10.1017/fms.2023.3  

   Miagkov, Konstantin; Thorne, Jack A. (January 2023, Forum of Mathematics, Sigma)

   Abstract Let F be a CM number field. We generalise existing automorphy lifting theorems for regular residually irreducible p -adic Galois representations over F by relaxing the big image assumption on the residual representation. 

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5. A New Family of Algebraically Defined Graphs with Small Automorphism Group

   https://doi.org/10.37236/10707  

   Lazebnik, Felix; Taranchuk, Vladislav (January 2022, The Electronic Journal of Combinatorics)

   Let $$p$$ be an odd  prime, $q=p^e$, $$e \geq 1$$, and $$\mathbb{F} = \mathbb{F}_q$$ denote the finite field of $$q$$ elements.  Let $$f: \mathbb{F}^2\to \mathbb{F}$$ and  $$g: \mathbb{F}^3\to \mathbb{F}$$  be functions, and  let $$P$$ and $$L$$ be two copies of the 3-dimensional vector space $$\mathbb{F}^3$$. Consider a bipartite graph $$\Gamma_\mathbb{F} (f, g)$$ with vertex partitions $$P$$ and $$L$$ and with edges defined as follows: for every $$(p)=(p_1,p_2,p_3)\in P$$ and every $$[l]= [l_1,l_2,l_3]\in L$$, $$\{(p), [l]\} = (p)[l]$$ is an edge in $$\Gamma_\mathbb{F} (f, g)$$ if $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = g(p_1,p_2,l_1).$$The following question  appeared in Nassau: Given $$\Gamma_\mathbb{F} (f, g)$$,  is it always possible to find a function $$h:\mathbb{F}^2\to \mathbb{F}$$ such that the graph $$\Gamma_\mathbb{F} (f, h)$$  with the same vertex set as $$\Gamma_\mathbb{F} (f, g)$$ and with edges $(p)[l]$  defined in a similar way  by the system $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = h(p_1,l_1),$$ is isomorphic to $$\Gamma_\mathbb{F} (f, g)$$ for infinitely many $$q$$?  In this paper we show that the  answer to the question is negative and the graphs $$\Gamma_{\mathbb{F}_p}(p_1\ell_1, p_1\ell_1p_2(p_1 + p_2 + p_1p_2))$$ provide such an example for $$p \equiv 1 \pmod{3}$$. Our argument is based on proving that the automorphism group of these graphs has order $$p$$, which is the smallest possible order of the automorphism group of graphs of the form $$\Gamma_{\mathbb{F}}(f, g)$$. 

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