2023 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 – 2025-03-31
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| Keywords | Local Hecke algebra / Local Newforms / Shimura correspondence |
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| Outline of Annual Research Achievements |
In a joint research with Moshe Baruch and Markos Karameris, we describe the subalgebra H of the metaplectic G:=\tilde{SL}_2(Q_p) of level p^n (with trivial and quadratic character) that is supported in the maximal compact K using generators and relations, in particular we show that it is a commutative 2n dimensional algebra and give a complex basis of eigenvectors under the left action of H on I(n) where the later is the induced representation from K_0(p^n) to K. In particular we obtain two level p^n new vectors with determined eigenaction. We consider principal series representations of G (induced from Borel of suitable characters of conductor p^n) and describe the 2-dimensional local newforms in these representations under the action of H.
In another work with Ramla Abdellatif, we describe the p-modular Iwahori-Hecke algebras (with trivial and quadratic) associated with G using generators and relations. We compute the center of these algebras and use this and the presentation to classify their finite dimensional simple modules, they are at most 2 dimensional. When the character is neither trivial nor quadratic, the p-modular Iwahori Hecke algebra turns out to be a polynomial algebra. In each of these cases, we compute the Iwahori-isotypical components of genuine principal series representations.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We plan to use the aforementiond results to compute local newforms in the automorphic representation of metaplectic SL_2(Q_p). We considered the case of irreducible principal series and will next consider the special and supercuspidals. We expect to obtain (partially) local newform theory of half integral weight for p odd. We have also done computations of subalgebra of level 2^n. We will return to this once we complete the odd p case.
On the p-modular Shimura correspondence work with Dr. Abdellatif, we not only recover and generalized Peskin’s results for arbitrary odd p but also classified the finite dimensional representations of Iwahori Hecke algebras (with character) and obtained the Iwahori invariant vectors of principal series representations.
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| Strategy for Future Research Activity |
In the complex representation setting, we next plan to consider the special representations of the metaplectic SL_2(Q_p) and compute the associated local newforms. The next goal will be to study the supercuspidal representations. By Shimura-Waldspurger, we know the correspondence between representations of metaplectic SL_2 and that of PGL_2. Using this and our computation of local newforms we hope to obtain a local newform theory of half integral weight for p odd. As mentioned above, we also plan to complete our computations of subalgebra of level 2^n and hope to obtain local newforms for p=2 case.
In p-modular setting, having described the Iwahori Hecke algebra and the Iwahori module structures, our next goal is to study and describe the connection with the pro-p Iwahori Hecke algebra and their modules. On the representation theoretic side, the goal is to determine the irreducible smooth representations of metaplectic SL_2(Q_p) (given Peskin's work and our generalization, we are left to describe the mod p genuine supersingular representation). This required study of the category of right modules over the Iwahori-Hecke algebras aforementioned. We plan to compare our classification results with those obtained by Witthaus for metaplectic GL_2 and hope to obtain some first instance of functoriality in the p-modular setting.
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| Causes of Carryover |
Due to teaching commitments, I have been unable to do research travel as I was intending to and this is the primary reason for not being able to use up the funds.
Usage plan for the next fiscal:
Research visit to Prof Moshe Baruch in Technion, Israel or host him at Tokyo Tech. Research visit to Dr. Ramla Abdellatif to complete the current article and start a new project. Local travel to attend conferences and present my research work.
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