{"id":3081,"date":"2025-12-11T21:26:25","date_gmt":"2025-12-11T19:26:25","guid":{"rendered":"https:\/\/mathematics.haifa.ac.il\/?p=3081"},"modified":"2025-12-18T10:49:48","modified_gmt":"2025-12-18T08:49:48","slug":"colloquium-tuesday-dec-16-2025-1400-room-614-speaker-yotam-hendel-ben-gurion-university-title-integrability-of-harish-chandra-characters-and-singularities-of-representations","status":"publish","type":"post","link":"https:\/\/mathematics.haifa.ac.il\/?p=3081","title":{"rendered":"Colloquium: Tuesday Dec. 16, 2025 (14:00, room 614). Speaker: Yotam Hendel (Ben-Gurion University). Title: &#8220;Integrability of Harish-Chandra characters and singularities of representations&#8221;."},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Abstract: Given a finite group G, and a finite dimensional complex representation \u03c0: G \\to GL(V), the character \u03c7<em>\u03c0(g):=tr(\u03c0(g)) of \u03c0 is a function on G which completely determines \u03c0. Let G = GL_n(F), where F is the field of real or p-adic numbers, or more generally let G be the group of F-points of a reductive algebraic group defined over a local field F of characteristic 0. In this setting, irreducible representations \u03c0 of G may be infinite dimensional. Nevertheless, one may define the Harish-Chandra character \u0398<\/em>\u03c0 of \u03c0, a distribution on G serving as the analogue of a finite-group character. A classical result due to Harish-Chandra asserts that \u0398<em>\u03c0 is regular, namely it is given by integration against a locally integrable function f<\/em>\u03c0 on G. In fact, f_\u03c0 has stronger integrability properties: it is locally L^{1+r}-integrable for some r&gt;0.<br>In recent joint work together with Itay Glazer and Julia Gordon, we define a new singularity invariant \u03f5<em>\u03c0 of a representation \u03c0, by considering the largest r such that f<\/em>\u03c0 is locally L^{1+r}-integrable. We explore \u03f5<em>\u03c0, show it is bounded from below only in terms of the group G, and calculate it in the case of p-adic GL_n. To do so, we relate \u03f5<\/em>\u03c0 to the integrability of certain natural measures on the Lie algebra of G arising from nilpotent matrices, and use the theory of hyperplane arrangements. We plan to discuss some of the above, as well as applications, as time permits.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Abstract: Given a finite group G, and a finite dimensional complex representation \u03c0: G \\to GL(V), the character \u03c7\u03c0(g):=tr(\u03c0(g)) of \u03c0 is a function on G which completely determines \u03c0. Let G = GL_n(F), where F is the field of&#8230;<br \/><a class=\"read-more-button\" href=\"https:\/\/mathematics.haifa.ac.il\/?p=3081\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-3081","post","type-post","status-publish","format-standard","hentry","category-colloquium"],"_links":{"self":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/3081","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3081"}],"version-history":[{"count":1,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/3081\/revisions"}],"predecessor-version":[{"id":3082,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/3081\/revisions\/3082"}],"wp:attachment":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3081"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3081"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3081"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}