2019 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 – 2022-03-31
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| Keywords | Hecke algebra / Whittaker functions / Automorphic / Representation |
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| Outline of Annual Research Achievements |
Let G=GL_2(Q_p) and K_0=K_0(p^n). In the Iwahori case (n=1), it is known that every irreducible finite dimensional representation of the Hecke algebra is at most two-dimensional; 1-parameter family of 1-dimensional representations of H corresponding to spherical representations of G, two 1-dimensional representation of H corresponding to Steinberg representations of G and the other characters of H corresponding to 1-dimensional representations of G (with trivial central character). The Whittaker function of the new vector in a spherical representation is described by Casselman-Shalika. We (together with Moshe Baruch) use Hecke algebra to give description of the Whittaker function of a new vector which is an Iwahori-fixed vector. Our computation matches with that of Scmidt, but differs in the method, more precisely we do not use realization of representation (Steinberg) to obtain the Whittaker function. The case n=2, p=2, we have a complete description of Hecke algebra and its center and using this we show that every irreducible finite dimensional representation of H(G//K_0(4)) modulo center is of dimension at most 3, these irreducible representations are given by one parameter family of three dimensional representation corresponding to the spherical representations, two 2-dimensional representations corresponding to the Steinberg representations and two characters corresponding to supercuspidal representations and characters of H corresponding to 1-dimensional representation of G. We give complete Whittaker function of the two supercuspidal 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
The description of full Hecke algebra and obtaining information purely from the presentations of algebra has been our general approach. In that, getting descriptions of complete Whittaker funtions corresponding to supercuspidals without using any realization of supercuspidals involved is important first step. The next step is to consider odd prime cases. We have obtained a description of full Hecke algebra for p=3 and n=2. In this case, one of the generators is “infinite” in the sense of not satisfying a finite algebraic equation. This corresponds to the fact that there are infinitely many principle series representations with p^2 new vector. The Hecke algebra of the double cover of SL_2 modulo the inverse image of K_0(p^2) intersection SL_2 should correspond, via local Shimura correspondence, to the Hecke algebra of PGL_2 modulo K_0(p^2) and so using the method illustrated above we expect a similar result for the complete Whittaker functions on the double cover of SL_2. This will be significant in computing the local factors in Baruch and Mao’s formula.
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| Strategy for Future Research Activity |
We plan to give complete description of local Hecke algebra of G modulo K_0(p^2). The support of these algebras increases as prime p increases and computations become intricate, however, since we do have a full description of Hecke subalgebra of functions that are supported on maxima compact subgroup, we obtain the values of Whittaker functions on specific double cosets. Although this is sufficient for the purpose of obtaining local factors in Baruch and Mao’s formula, nevertheless, it will be interesting to obtain complete Whittaker functions. The left out double cosets on which we are unable to describe the Whittaker functions just using the subalgebra, provides us with candidates of generators for the full Hecke algebra. In the case p=3, n=2, using this trick, we could reduce our attention to the generator coming from the double coset of y(p)w(p^n)y(p) which turned out to be the infinite generator. We expect similar situation for general odd prime. The novelty of our approach to computation of full Whittaker function of new vectors is that we do not need to use any realization of representations, on the other hand, a Whittaker function does determine the representation completely and one can use it to describe the action on the Kirillov model. To describe the finite dimensional representations of Hecke algebra H(G//K_0(4)), we used the central elements of the Hecke algebra. For general odd primes, we therefore plan to obtain generators of the center. The next step is to deal with the analogue cases for the double cover of SL_2 and obtain complete Whittaker functions.
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| Causes of Carryover |
The usage of budget is lesser than expected due to travel restrictions imposed by ongoing coronavirus pandemic and teaching obligations. Further, limited travels that I could do or travel money that I used to invite my collaborator were funded by my current institute and host institutes.
I am planning to organize a number theory workshop during Spring 2021 (depending on the pandemic situation) and I intend to use my budget for the same.
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