2022 Fiscal Year Research-status Report
Newform theory for the full space via local Shimura correspondence and Waldspurger-type theorem
| Project/Area Number |
18K13396
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
プルカイト ソーマ 東京工業大学, 理学院, 特任准教授 (30806592)
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | Whittaker functions / Iwahori Hecke algebra / mod p representation |
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| Outline of Annual Research Achievements |
Shintani gave an explicit formula for spherical Whittaker function associated to unramified principal series representation of GL_n(F) where F is a non-archimedean local field of characteristic zero. Miyauchi extends work of Shintani and computes Whittaker functions associated to newforms (fixed by certain K_c, defined by Jaquet and Shalika) for GL_n(F). The space of such newforms is 1-dimensional and Miyauchi uses Hecke algebra associated to K_c to give description of associated Whittaker function on BK_c (subgroup smaller than GL_n). In a similar spirit, we consider Whittaker functions associated to Iwahori subgroup J of GL_n(F). The Hecke algebra modulo K_c is bigger than the Hecke algebra modulo J. Assuming that J-fixed space is one-dimensional, we attempt to describe Whittaker function on full GL_n(F). There is a bijection between the irreducible representations of GL_n containing an Iwahori fixed vector and irreducible finite dimensional representations of the Iwahori Hecke algebra. Our Whittaker function are associated to representations that correspond to one-dimensional representations of the Iwahori Hecke algebra. This is joint with Moshe Baruch and Markos Karameris.
In another project with Ramla Abdellatif, we consider mod p genuine Hecke algebras associated to metaplectic 2-cover of SL_2(Q_p). Laura Peskin describes the spherical ones in the cases p=1 mod(4) and obtains Savin-type local Shimura correspondence in this setting. We obtain such correspondence for all odd primes.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
Our current research is slightly drifted from the original goal. We have been interested in complete description of Whittaker functions using Hecke algebras. We were able to do this for Whittaker new vectors of GL_2(Q_p) of level p^2 by describing Hecke algebra of GL_2(Q_p) modulo K_0(p^2) for any prime p. This seems difficult for higher power p^n, although there is a hope when p=2. While studying this, we found that even for the Iwahori case for GL_n for n>2, there is no full description of Whittaker function, although we have a beautiful description of the Hecke algebra itself. So, we would like to first consider this question.
Regarding local Shimura correspondence, there has been recent works of Peskin (2-cover of SL_2/Iwahori) and Witthaus (2-cover of GL_2/pro-p Iwahori) in the mod-p setting, p odd. Their work leads to certain questions (as described in the future plan) that we would like to study.
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| Strategy for Future Research Activity |
On the topic of Whittaker function of GL_n, we plan to study associated representation theoretic questions. We also plan to come back to our question of description of the Hecke algebra of G=GL_2(Q_p) modulo K_0=K_0(2^n) for any n and then obtain corresponding presentation for double cover Hecke algebra and compute associated Whittaker new vector of level 2^n. We have description of the subalgebra that are supported on K. In the metaplectic setting, we plan to start with describing such subalgebras.
In the mod-p setting, Peskin obtained local Shimura correspondence (isomorphism) between mod p genuine spherical Hecke algebra of 2-cover of SL_2(Q_p) and spherical Hecke algebra of its reductive dual group PGL_2 for p=1(mod 4). We extended this for all odd primes p. Peskin also showed that there is no longer an isomorphism when considering Iwahori Hecke algebra (instead of spherical); different from characteristic 0 case by Savin. We plan to consider the Iwahori case for all odd primes p. Peskin described irreducible smooth genuine ordinary mod p reps. of 2-cover of SL_2(Q_p), Witthaus describes all such (incuding supersingular) mod p reps. of 2-cover of GL_2(Q_p). In joint with Abdellatif, we plan to study and describe mod p genuine supersingular reps. of 2-cover of SL_2(Q_p) and compare it with restriction from Witthaus’s work on 2-cover of GL_2(Q_p). If p=2, no genuine mod 2 reps. of 2-cover of SL_2(Q_2) as mu_2 acts trivially. But one can consider mod l (l not 2) reps., we would like to consider this question in the context of Savin’s local Shimura correspondence.
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| Causes of Carryover |
Travel plan for reserach collaboration:
1. Visit Prof. Moshe Baruch (Technion, Israel) ~ 500,000 JPY (flight + 1-week stay)
2. Visit Prof. Ramla Abdellatiff (UPJV, France) ~ 600,000 JPY (flight + 1-week stay)
3. Visit Prof. Jerome Dimabayao (University of the Philippines) ~ 200,000 JPY (flight + 1-week stay)
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