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SUMMARY:Test vectors for some ramified representations of $GL(2)$
DTSTART:20180522T151500
DTEND:20180522T161500
DTSTAMP:20260406T055935Z
UID:d91ff82d44f14999cf02c947b65dd1abb35322aae379eb0d020eed34
CATEGORIES:Conferences - Seminars
DESCRIPTION:Vinayak Vatsal (University of British Columbia)\nAbstract: We 
 give an explicit construction of test vectors for $T$-equivariant linear f
 unctionals on representations $\\Pi$ of $GL_2$ of a local field\, where $T
 $ is a non-split torus. Of particular interest is the case when both the r
 epresentations are ramified\; we completely solve this problem for princip
 al series and Steinberg representations of $GL_2$\, as well as for depth z
 ero supercuspidals over $\\qq_p$. A key ingredient is a theorem of Casselm
 an and Sillberger\, which allows us to quickly reduce almost all cases to 
 that of the principal series\, which can be analyzed directly. Our method 
 shows that the only genuinely difficult cases are the characters of $T$ wh
 ich occur in the primitive part (or ``type") of $\\Pi$ when $\\Pi$ is supe
 rcuspidal\, and we resolve the depth zero case. The method to handle the d
 epth zero case is based on modular representation theory\, motivated by co
 nsiderations from Deligne-Lusztig theory and the de Rham cohomology of Del
 igne-Lusztig-Drinfeld curves. The proof also reveals some interesting feat
 ures related to the Langlands correspondence in characteristic $p$. We sho
 w in particular that the test vector problem has an obstruction in charact
 eristic $p$ beyond the root number criterion of Waldspurger and Tunnell\, 
 and exhibits an unexpected dichotomy related to the weights in Serre's con
 jecture and the signs of standard Gauss sums.
LOCATION:PH H3 31 https://plan.epfl.ch/?room==PH%20H3%2031
STATUS:CONFIRMED
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