Bessel functions and Kloosterman integrals on \(\mathrm{GL}(n)\). (English) Zbl 07796915
The classical Bessel functions are solutions to the differential equation
\[
x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0,
\]
where \(\alpha \in \mathbb{C}\) is a fixed complex parameter. They appear naturally in various questions related to automorphic representations of \(\mathrm{GL}_2\), most notably, in summation formulas such as the Voronoi summation formula or the Kuznetsov trace formula.
Given the ubiquity of trace formula methods in the theory of automorphic forms, it is desirable to generalize the definition of Bessel functions and attach them to representations of \(\mathrm{GL}_n\) for \(n>2\). Over \(\mathrm{GL}_n(F)\), where \(F\) is a \(p\)-adic field, this has been done by E. M. Baruch [Ohio State Univ. Math. Res. Inst. Publ. 11, 1–40 (2005; Zbl 1103.22006)]. A key technical property of Bessel functions is that of local integrability (this is analogous to the local integrablity of Harish-Chandra’s character distribution). For \(\mathrm{GL}_2\) and \(\mathrm{GL}_3\) this is a result due to Baruch. In the paper under review, the author proves the local integrability of Bessel functions for \(\mathrm{GL}_n(\mathbb{Q}_p)\), \(n\geq 4\).
In order to state the main result, we briefly sketch the definition of Bessel functions for \(G=\mathrm{GL}_n(F)\). Let \(B=TN\) be the standard Borel subgroup of \(G\): \(B\) is the subgroup consisting of all upper-triangular matrices in \(G\), \(T\) is the diagonal torus, and \(N\) is the unipotent radical (i.e.the subgroup of \(G\) consisting of upper triangular matrices with \(1\)’s on the diagonal). The decomposition of \(G\) into double cosets of \(B\) is known as the Bruhat decomposition; the open cell in this decomposition is \(BwB\), where \(w \in G\) is the element with \(1\)’s on the anti-diagonal and \(0\)’s elsewhere.
Fix a generic character of \(\psi\) of \(N\). Then it is well known that – given any irreducible (admissible) representation \((\pi,V)\) of \(G\) – the space of functionals \(L\) on \(V\) satisfying \[ L(\pi(n)v) = \psi(n)L(v) \] is at most one-dimensional. A non-zero \(L\) as above is called a Whittaker functional; a representation which possesses a Whittaker functional is said to be generic.
Suppose now that \(\pi\) is generic and fix a Whittaker functional \(L\) for \(\pi\). Given a vector \(v \in V\), let \(W_v: G\to \mathbb{C}\) be the function defined by \(W_v(g) = L(\pi(g)v)\). Then \(v \mapsto W_v\) is an isomorphism of \((\pi,V)\) with the space \[ \mathcal{W}(\pi,\psi) = \{W: G \to \mathbb{C}: W(ng) = \psi(n)W(g)\} \] on which \(G\) acts by right translation. We call \(\mathcal{W}(\pi,\psi)\) the Whittaker model of \((\pi,V)\).
Fix an element \(W \in \mathcal{W}(\pi,\psi)\). Let \(N_1 \subseteq N_2 \subseteq N_3 \subseteq \dotsb\) be a filtration of \(N\) by open compact subgroups. D. Soudry [Duke Math. J. 51, 355–394 (1984; Zbl 0557.12012)] showed that the limit \[ \lim_{k \to \infty }\int_{N_k} W(gn) \psi^{-1}(n) dn \] exists for any \(g \in BwB\), and is independent of the choice of the filtration. This leads to another Whittaker functional, \(L_g\), defined by \[ L_g(v) = \lim_{k \to \infty }\int_{N_k} W_v(gn) \psi^{-1}(n) dn. \] By the uniqueness of Whittaker functionals, there exists a scalar \(j_{\pi,\psi}(g)\) such that \[ L_g(v)= j_{\pi,\psi}(g)L(v) \quad \text{for all } v \in V. \] We have thus obtained a function \(j_{\pi,\psi}\) defined on the open Bruhat cell \(BwB\), which we extend to all of \(G\) by setting \(j_{\pi,\psi} \equiv 0\) outside of \(BwB\). The function \(j_{\pi,\psi}\) is called the Bessel function attached to \(\pi\). The main result of this paper is the following:
Theorem (Theorem 2.4). Let \(\pi\) be a generic representation of \(\mathrm{GL}_n(\mathbb{Q}_p)\). Then the Bessel function \(j_{\pi,\psi}\) is locally integrable.
The result is obtained by reducing the question of local integrability to the problem of finding a non-trivial upper bound for Kloosterman sums for \(\mathrm{GL}_n\). This is in turn accomplished using a method developed by G. Stevens [Math. Ann. 277, 25–51 (1987; Zbl 0597.12017)].
Given the ubiquity of trace formula methods in the theory of automorphic forms, it is desirable to generalize the definition of Bessel functions and attach them to representations of \(\mathrm{GL}_n\) for \(n>2\). Over \(\mathrm{GL}_n(F)\), where \(F\) is a \(p\)-adic field, this has been done by E. M. Baruch [Ohio State Univ. Math. Res. Inst. Publ. 11, 1–40 (2005; Zbl 1103.22006)]. A key technical property of Bessel functions is that of local integrability (this is analogous to the local integrablity of Harish-Chandra’s character distribution). For \(\mathrm{GL}_2\) and \(\mathrm{GL}_3\) this is a result due to Baruch. In the paper under review, the author proves the local integrability of Bessel functions for \(\mathrm{GL}_n(\mathbb{Q}_p)\), \(n\geq 4\).
In order to state the main result, we briefly sketch the definition of Bessel functions for \(G=\mathrm{GL}_n(F)\). Let \(B=TN\) be the standard Borel subgroup of \(G\): \(B\) is the subgroup consisting of all upper-triangular matrices in \(G\), \(T\) is the diagonal torus, and \(N\) is the unipotent radical (i.e.the subgroup of \(G\) consisting of upper triangular matrices with \(1\)’s on the diagonal). The decomposition of \(G\) into double cosets of \(B\) is known as the Bruhat decomposition; the open cell in this decomposition is \(BwB\), where \(w \in G\) is the element with \(1\)’s on the anti-diagonal and \(0\)’s elsewhere.
Fix a generic character of \(\psi\) of \(N\). Then it is well known that – given any irreducible (admissible) representation \((\pi,V)\) of \(G\) – the space of functionals \(L\) on \(V\) satisfying \[ L(\pi(n)v) = \psi(n)L(v) \] is at most one-dimensional. A non-zero \(L\) as above is called a Whittaker functional; a representation which possesses a Whittaker functional is said to be generic.
Suppose now that \(\pi\) is generic and fix a Whittaker functional \(L\) for \(\pi\). Given a vector \(v \in V\), let \(W_v: G\to \mathbb{C}\) be the function defined by \(W_v(g) = L(\pi(g)v)\). Then \(v \mapsto W_v\) is an isomorphism of \((\pi,V)\) with the space \[ \mathcal{W}(\pi,\psi) = \{W: G \to \mathbb{C}: W(ng) = \psi(n)W(g)\} \] on which \(G\) acts by right translation. We call \(\mathcal{W}(\pi,\psi)\) the Whittaker model of \((\pi,V)\).
Fix an element \(W \in \mathcal{W}(\pi,\psi)\). Let \(N_1 \subseteq N_2 \subseteq N_3 \subseteq \dotsb\) be a filtration of \(N\) by open compact subgroups. D. Soudry [Duke Math. J. 51, 355–394 (1984; Zbl 0557.12012)] showed that the limit \[ \lim_{k \to \infty }\int_{N_k} W(gn) \psi^{-1}(n) dn \] exists for any \(g \in BwB\), and is independent of the choice of the filtration. This leads to another Whittaker functional, \(L_g\), defined by \[ L_g(v) = \lim_{k \to \infty }\int_{N_k} W_v(gn) \psi^{-1}(n) dn. \] By the uniqueness of Whittaker functionals, there exists a scalar \(j_{\pi,\psi}(g)\) such that \[ L_g(v)= j_{\pi,\psi}(g)L(v) \quad \text{for all } v \in V. \] We have thus obtained a function \(j_{\pi,\psi}\) defined on the open Bruhat cell \(BwB\), which we extend to all of \(G\) by setting \(j_{\pi,\psi} \equiv 0\) outside of \(BwB\). The function \(j_{\pi,\psi}\) is called the Bessel function attached to \(\pi\). The main result of this paper is the following:
Theorem (Theorem 2.4). Let \(\pi\) be a generic representation of \(\mathrm{GL}_n(\mathbb{Q}_p)\). Then the Bessel function \(j_{\pi,\psi}\) is locally integrable.
The result is obtained by reducing the question of local integrability to the problem of finding a non-trivial upper bound for Kloosterman sums for \(\mathrm{GL}_n\). This is in turn accomplished using a method developed by G. Stevens [Math. Ann. 277, 25–51 (1987; Zbl 0597.12017)].
Reviewer: Petar Bakić (Salt Lake City)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11F85 | \(p\)-adic theory, local fields |
References:
| [1] | Aizenbud, A.; Gourevitch, D.; Komarsky, A., Vanishing of certain equivariant distributions on \(p\)-adic spherical spaces, and nonvanishing of spherical Bessel functions. Int. Math. Res. Not., 18, 8471-8483 (2015) · Zbl 1330.22021 |
| [2] | Aizenbud, A.; Gourevitch, D.; Sayag, E., \(z\)-finite distributions on \(p\)-adic groups. Adv. Math., 1376-1414 (2012) · Zbl 1327.22012 |
| [3] | Baruch, E. M., On Bessel distributions of \(\operatorname{GL}(2)\) over a \(p\)-adic field. J. Number Theory, 190-202 (1997) · Zbl 0887.22023 |
| [4] | Baruch, E. M., On Bessel distributions for quasi-split groups. Trans. Am. Math. Soc., 7, 2601-2614 (2001) · Zbl 0976.22015 |
| [5] | Baruch, E. M., Bessel functions for \(\operatorname{GL}(3)\) over a \(p\)-adic field. Pac. J. Math., 1, 1-34 (2003) |
| [6] | Baruch, E. M., Bessel distribution for \(\operatorname{GL}(3)\) over the \(p\)-adic field. Pac. J. Math., 1, 11-27 (2004) · Zbl 1064.22008 |
| [7] | Baruch, E. M., Bessel functions for \(\operatorname{GL}(n)\) over a \(p\)-adic field. Automorphic representations, 1-40 |
| [8] | Bump, D.; Friedberg, S.; Goldfeld, D., Poincare series and Kloosterman sums for \(\operatorname{SL}(3, \mathbb{Z})\). Acta Arith., 31-89 (1988) · Zbl 0647.10020 |
| [9] | Blomer, V., Density theorems for \(\operatorname{GL}(n)\). Invent. Math., 783-811 (2023) |
| [10] | Blomer, V.; Man, S., Bounds for Kloosterman sums on \(\operatorname{GL}(n) (2022)\), to appear in Math. Ann. |
| [11] | Chai, J., A weak kernel formula for Bessel functions. Trans. Am. Math. Soc., 10, 7139-7167 (2017) · Zbl 1388.22011 |
| [12] | Chai, J., On Bessel functions over \(p\)-adic fields. Int. Math. Res. Not., 3, 673-699 (2019) · Zbl 1428.33044 |
| [13] | Cogdell, J., Bessel functions for \(\operatorname{GL}_2\). Indian J. Pure Appl. Math., 5, 557-582 (2014) · Zbl 1365.11060 |
| [14] | Dabrowski, R.; Fisher, B., A stationary phase formula for exponential sums over \(\mathbb{Z} / p^m \mathbb{Z}\) and applications to \(\operatorname{GL}(3)\)-Kloosterman sums. Acta Arith., 1, 1-48 (1997) · Zbl 0893.11032 |
| [15] | Dabrowski, R.; Reeder, M., Kloosterman sets in reductive groups. J. Number Theory, 2, 228-255 (1998) · Zbl 0919.11055 |
| [16] | Feigon, B.; Lapid, E.; Offen, O., On representations distinguished by unitary groups. Publ. Math. Inst. Hautes Études Sci., 185-323 (2012) · Zbl 1329.11053 |
| [17] | Friedberg, S., Poincare series for \(\operatorname{GL}(n)\): Fourier expansion, Kloosterman sums and algebreo-geometric estimates. Math. Z., 165-188 (1987) · Zbl 0612.10020 |
| [18] | Goldfeld, D.; Stade, E.; Woodbury, M., An orthogonality relation for \(\operatorname{GL}(4, \mathbb{R})\), 1-83, with an appendix by Bingrong Huang |
| [19] | Harish-Chandra, Harmonic Analysis on Reductive \(p\)-adic Groups. Lecture Notes in Mathematics (1970), Springer-Verlag: Springer-Verlag Berlin, notes by G. van Dijk · Zbl 0202.41101 |
| [20] | Harish-Chandra, Admissible Invariant Distributions on Reductive \(p\)-adic Groups. University Lecture Series (1999), American Mathematical Society: American Mathematical Society Providence, RI, preface and notes by Stephen DaBacker and Paul J. Sally, Jr. · Zbl 0928.22017 |
| [21] | Ichino, A.; Templier, N., On the Voronoi formula for \(\operatorname{GL}(n)\). Am. J. Math., 1, 65-101 (2013) · Zbl 1284.11077 |
| [22] | Jacquet, H., Smooth transfer of Kloosterman integrals. Duke Math. J., 1, 121-152 (2003) · Zbl 1060.11032 |
| [23] | Jacquet, H., Kloosterman identities over a quadratic extension. Ann. Math., 2, 755-779 (2004) · Zbl 1071.11026 |
| [24] | Jacquet, H.; Ye, Y., Relative Kloosterman integrals for \(\operatorname{GL}(3)\). Bull. Soc. Math. Fr., 263-295 (1992) · Zbl 0785.11032 |
| [25] | Jacquet, H.; Ye, Y., Distinguished representation and quadratic base change for \(\operatorname{GL}(3)\). Trans. Am. Math. Soc., 3, 913-939 (1996) · Zbl 0861.11033 |
| [26] | Jacquet, H.; Ye, Y., Germs of Kloosterman integrals for \(\operatorname{GL}(3)\). Trans. Am. Math. Soc., 3, 1227-1255 (1999) · Zbl 0919.11038 |
| [27] | Knightly, A.; Li, C., Kuznetsov’s Trace Formula and the Hecke Eigenvalues of Maass Forms. Memoirs American Math. Society (2013), Number 1055 · Zbl 1314.11038 |
| [28] | Kiral, E. Mehmet; Nakasuji, M., Parametrization of Kloosterman sets and \(S L_3\)-Kloosterman sums. Adv. Math. (2022) · Zbl 1496.11106 |
| [29] | Lapid, E.; Mao, Z., Stability of certain oscillatory integrals. Int. Math. Res. Not., 3, 525-547 (2013) · Zbl 1388.22015 |
| [30] | Man, S., Symplectic Kloosterman sums and Poincare series. Ramanujan J., 707-753 (2022) · Zbl 1489.11114 |
| [31] | Qi, Z., Theory of Fundamental Bessel Functions of High Rank. Memoirs American Math. Society (2020), Number 1303 · Zbl 1464.33001 |
| [32] | Sarnak, P., Diophantine problems and linear groups, 459-471 · Zbl 0743.11018 |
| [33] | Shalika, J., The multiplicity one theorem for \(\operatorname{GL}(n)\). Ann. Math., 2, 171-193 (1974) · Zbl 0316.12010 |
| [34] | Soudry, D., The \(L\) and \(γ\) factors for generic representations of \(\operatorname{GSp}_4 \times \operatorname{GL}_2\) over a non-Archimedean \(p\)-adic field. Duke Math. J., 355-394 (1984) · Zbl 0557.12012 |
| [35] | Stevens, G., Poincare series on \(\operatorname{GL}(r)\) and Kloostermann sums. Math. Ann., 25-51 (1987) · Zbl 0597.12017 |
| [36] | Watson, G., A Treatise on the Theory of Bessel Functions (1995), Cambridge University Press · Zbl 0849.33001 |
| [37] | Weil, A., On some exponential sums. Proc. Natl. Acad. Sci. USA, 204-207 (1948) · Zbl 0032.26102 |
| [38] | Ye, Y., The fundamental lemma of a relative trace formula for \(\operatorname{GL}(3)\). Compos. Math., 121-162 (1993) · Zbl 0799.11013 |
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