| 研究課題/領域番号 |
18K13396
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| 研究種目 |
若手研究
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| 配分区分 | 基金 |
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| 審査区分 |
小区分11010:代数学関連
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| 研究機関 | 東京工業大学 |
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研究代表者 |
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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| 研究期間 (年度) |
2018-04-01 – 2025-03-31
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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
4,290千円 (直接経費: 3,300千円、間接経費: 990千円)
2021年度: 1,040千円 (直接経費: 800千円、間接経費: 240千円)
2020年度: 1,040千円 (直接経費: 800千円、間接経費: 240千円)
2019年度: 1,040千円 (直接経費: 800千円、間接経費: 240千円)
2018年度: 1,170千円 (直接経費: 900千円、間接経費: 270千円)
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| キーワード | Local Hecke algebra / Local Newforms / Shimura correspondence / Whittaker functions / Iwahori Hecke algebra / mod p representation / Hecke algebra / Congruent numbers / Automorphic / Representation / Automorphic forms / Half-integral weight / Newforms / Waldspurger's formula |
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| 研究実績の概要 |
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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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
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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| 今後の研究の推進方策 |
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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