Rigid inner forms over local function fields

We generalize the concept of rigid inner forms, defined by Kaletha in [Kal16], to the setting of a local function field $F$ in order state the local Langlands conjectures for arbitrary connected reductive groups over $F$. To do this, we define for a connected reductive group $G$ over $F$ a new cohomology set $H^{1}(\mathcal{E}, Z \to G) \subset H_{\text{fpqc}}^{1}(\mathcal{E}, G)$ for a gerbe $\mathcal{E}$ attached to a class in $H_{\text{fppf}}^{2}(F, u)$ for a certain canonically-defined profinite commutative group scheme $u$, building up to an analogue of the classical Tate-Nakayama duality theorem. We define a relative transfer factor for an endoscopic datum serving a connected reductive group $G$ over $F$, and use rigid inner forms to extend this to an absolute transfer factor, enabling the statement of endoscopic conjectures relating stable virtual characters and $\dot{s}$-stable virtual characters for a semisimple $\dot{s}$ associated to a tempered Langlands parameter.Comment: v3: submitted version, significant expository changes, 77 page

Rigid inner forms over global function fields

We construct an fpqc gerbe $\mathcal{E}_{\dot{V}}$ over a global function field $F$ such that for a connected reductive group $G$ over $F$ with finite central subgroup $Z$, the set of $G_{\mathcal{E}_{\dot{V}}}$-torsors contains a subset $H^{1}(\mathcal{E}_{\dot{V}}, Z \to G)$ which allows one to define a global notion of ($Z$-)rigid inner forms. There is a localization map $H^{1}(\mathcal{E}_{\dot{V}}, Z \to G) \to H^{1}(\mathcal{E}_{v}, Z \to G)$, where the latter parametrizes local rigid inner forms (cf. [Kal16, Dil23]) which allows us organize local rigid inner forms across all places $v$ into coherent families. Doing so enables a construction of (conjectural) global $L$-packets and a conjectural formula for the multiplicity of an automorphic representation $\pi$ in the discrete spectrum of $G$ in terms of these $L$-packets. We also show that, for a connected reductive group $G$ over a global function field $F$, the adelic transfer factor $\Delta_{\mathbb{A}}$ for the ring of adeles $\mathbb{A}$ of $F$ serving an endoscopic datum for $G$ decomposes as the product of the normalized local transfer factors from [Dil20].Comment: Exposition heavily edited for clarity and brevity. Fixed minor errors related to definitions of the groups P_E,S,n and computations of derived inverse limits, and various typos. 63 pages, submitted versio

Minimal Length Maximal Green Sequences and Triangulations of Polygons

We use combinatorics of quivers and the corresponding surfaces to study maximal green sequences of minimal length for quivers of type $\mathbb{A}$. We prove that such sequences have length $n+t$, where $n$ is the number of vertices and $t$ is the number of 3-cycles in the quiver. Moreover, we develop a procedure that yields these minimal length maximal green sequences.Comment: 22 pages, 1 figur

Rigid Inner Forms Over Function Fields

We generalize the concept of rigid inner forms, defined by Kaletha in [Kal16] and [Kal18], to the setting of a local or global function field F in order to study endoscopy over F and state conjectures regarding representations of an arbitrary connected reductive group G over F . To do this, we define for such G a new cohomology set H1(E,Z → G) ⊂ H1 (E,G), where E is an fpqc A-gerbe over F attached to a class in H2 (F,A) for an explicit profinite commutative fppf group scheme A depending only on F (not on G), and extend the classical Tate-Nakayama duality theorem (locally), Tate’s global duality (cf. [Tat66]) result for tori, and their reductive analogues to these new expanded cohomology sets. We define a relative transfer factor for an endoscopic datum serving a connected reductive group G over local F , and use rigid inner forms to extend this to an absolute transfer factor, enabling the statement of endoscopic conjectures relating stable virtual characters and s ̇-stable virtual characters for a semisimple s ̇ associated to a tempered (local) Langlands parameter. Using global rigid inner forms, a localization map from the local gerbe to its global counterpart allows us to organize sets of local rigid inner forms into coherent families, allowing for a definition of global L-packets and a conjectural formula for the multiplicity of an automorphic representation π in the discrete spectrum of G in terms of these L-packets. We also show that, for a connected reductive group G over a global function field F , the adelic transfer factor ∆A for the ring of adeles A of global F serving an endoscopic datum for G decomposes as the product of the normalized local transfer factors.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/174549/1/dillery_1.pd