Bessel functions outside GL p 2 q . Jack Buttcane University of Maine Orono, ME 26 September 2020 Jack Buttcane Bessel functions outside GL p 2 q . 1/14
What are Bessel functions outside of GL p 2 q ? Let G “ PGL p n , R q “ GL p n , R q{ R ˆ , U p R q the upper triangular unipotent matrices, Y the diagonal matrices, Y ` the positive diagonal matrices, K “ PO p n , R q . Jack Buttcane Bessel functions outside GL p 2 q . 2/14
What are Bessel functions outside of GL p 2 q ? Define characters of Y : ˆ a 1 ˙ ˆ a 1 ˙ n n ź ź ... | a i | µ i , ... sgn p a i q δ i , p µ “ χ δ “ a n a n i “ 1 i “ 1 Jack Buttcane Bessel functions outside GL p 2 q . 3/14
What are Bessel functions outside of GL p 2 q ? Define characters of Y : ˆ a 1 ˙ ˆ a 1 ˙ n n ź ź ... | a i | µ i , ... sgn p a i q δ i , p µ “ χ δ “ a n a n i “ 1 i “ 1 of U p R q : ψ y p x q “ ψ I p yxy ´ 1 q “ e p y 1 x 1 ` . . . ` y n ´ 1 x n ´ 1 q , ¨ ˛ ¨ ˛ y 1 ¨¨¨ y n ´ 1 y 1 ¨¨¨ y n ´ 2 1 x n ˚ ˚ ‹ ... ... ... ˝ ‚ y “ , x “ , ˝ ‚ 1 x 1 y 1 1 1 Jack Buttcane Bessel functions outside GL p 2 q . 3/14
What are Bessel functions outside of GL p 2 q ? Define characters of Y : ˆ a 1 ˙ ˆ a 1 ˙ n n ź ź ... | a i | µ i , ... sgn p a i q δ i , p µ “ χ δ “ a n a n i “ 1 i “ 1 of U p R q : ψ y p x q “ ψ I p yxy ´ 1 q “ e p y 1 x 1 ` . . . ` y n ´ 1 x n ´ 1 q , ¨ ˛ ¨ ˛ y 1 ¨¨¨ y n ´ 1 y 1 ¨¨¨ y n ´ 2 1 x n ˚ ˚ ‹ ... ... ... ˝ ‚ y “ , x “ , ˝ ‚ 1 x 1 y 1 1 1 normalizations: ρ “ p n ´ 1 2 , n ´ 3 2 , . . . , ´ n ´ 1 2 q , use p ρ ` µ , assume ÿ n µ i “ 0, extend to G by p ρ ` µ p xyk q “ p ρ ` µ p y q . i “ 1 Jack Buttcane Bessel functions outside GL p 2 q . 3/14
What are Bessel functions outside of GL p 2 q ? Spherical Jacquet-Whittaker function: ż ˆ ˙ 1 ... W p g , µ, ψ q “ p ρ ` µ p w l xg q ψ p x q dx , w l “ . 1 U p R q Jack Buttcane Bessel functions outside GL p 2 q . 4/14
What are Bessel functions outside of GL p 2 q ? Spherical Jacquet-Whittaker function: ż ˆ ˙ 1 ... W p g , µ, ψ q “ p ρ ` µ p w l xg q ψ p x q dx , w l “ . 1 U p R q Since p ρ ` µ is an eigenfunction of the Casimir operators, so is the Whittaker function, with the same eigenvalues. Call µ the (Harish-Chandra/Langlands) spectral parameters. Jack Buttcane Bessel functions outside GL p 2 q . 4/14
What are Bessel functions outside of GL p 2 q ? Conjecture (Interchange of Integrals) If f p µ q is homorphic with rapid decay on an open tube domain containing Re p µ q “ 0, and y P Y , t P G , then ż ż f p µ q W p yw l xt , µ, ψ I q d µ ψ I p x q dx U p R q Re p µ q“ 0 ż f p µ q r “ K w l p y , t , µ q d µ, Re p µ q“ 0 where ż ´ ¯ r } x } K w l p y , t , µ q : “ lim h W p yw l xt , µ, ψ I q ψ I p x q dx R R Ñ8 U p R q is smooth in t and y , entire in µ and polynomially bounded in the coordinates of y , t , y ´ 1 , t ´ 1 , µ for Re p µ q in some fixed, compact set and h smooth and compactly supported with h p 0 q “ 1. Convergence! Jack Buttcane Bessel functions outside GL p 2 q . 5/14
What are Bessel functions outside of GL p 2 q ? It follows from work of Shalika that r K w l p y , t , µ q “ K w l p y , µ q W p t , µ, ψ I q for some function K w l p y , µ q , and this is called the long-element, spherical Bessel function for GL p n q . Jack Buttcane Bessel functions outside GL p 2 q . 6/14
What are Bessel functions outside of GL p 2 q ? It follows from work of Shalika that r K w l p y , t , µ q “ K w l p y , µ q W p t , µ, ψ I q for some function K w l p y , µ q , and this is called the long-element, spherical Bessel function for GL p n q . K w l p y , µ q has the bi- U p R q -invariance property K w l p xy , µ q “ K w l p y p w l xw l q , µ q “ ψ I p x q K w l p y , µ q , and is an eigenfunction of the Casimir operators with eigenvalues matching p ρ ` µ . Jack Buttcane Bessel functions outside GL p 2 q . 6/14
What are Bessel functions outside of GL p 2 q ? It follows from work of Shalika that r K w l p y , t , µ q “ K w l p y , µ q W p t , µ, ψ I q for some function K w l p y , µ q , and this is called the long-element, spherical Bessel function for GL p n q . K w l p y , µ q has the bi- U p R q -invariance property K w l p xy , µ q “ K w l p y p w l xw l q , µ q “ ψ I p x q K w l p y , µ q , and is an eigenfunction of the Casimir operators with eigenvalues matching p ρ ` µ . There are Bessel functions K w p y , µ, δ q attached to the other elements of the Weyl group and to non-spherical χ δ ‰ 1, as well. Jack Buttcane Bessel functions outside GL p 2 q . 6/14
What do we know about these Bessel functions? Conjecture (Asymptotics) Dropping the µ integral and ignoring issues of convergence, replacing the Whittaker function in the definition of K w p y , µ, δ q with its first-term asymptotics as Y Q y Ñ 0 yields the first-term asymptotics of K w p y , µ, δ q . Jack Buttcane Bessel functions outside GL p 2 q . 7/14
What do we know about these Bessel functions? Conjecture (Asymptotics) Dropping the µ integral and ignoring issues of convergence, replacing the Whittaker function in the definition of K w p y , µ, δ q with its first-term asymptotics as Y Q y Ñ 0 yields the first-term asymptotics of K w p y , µ, δ q . Conjecture (Uniqueness) The function K w p y , µ, δ q is uniquely determined by its first-term asymptotics, bi- U p R q -invariance and eigenvalues under the Casimir operators. Jack Buttcane Bessel functions outside GL p 2 q . 7/14
What do we know about these Bessel functions? The uniqueness conjecture is true in the long element case: p ´ ρ p g q K w l p gg T , µ, δ q is the spherical Whittaker function W p g , 2 µ, ψ 2 I q , up to a function C p µ, δ q Jack Buttcane Bessel functions outside GL p 2 q . 8/14
What do we know about these Bessel functions? The uniqueness conjecture is true in the long element case: p ´ ρ p g q K w l p gg T , µ, δ q is the spherical Whittaker function W p g , 2 µ, ψ 2 I q , up to a function C p µ, δ q so for y P Y ` , K w l p y , µ, δ q “ C p µ, δ q p ρ { 2 p y q W p? y , 2 µ, ψ 2 I q Jack Buttcane Bessel functions outside GL p 2 q . 8/14
What do we know about these Bessel functions? The uniqueness conjecture is true in the long element case: p ´ ρ p g q K w l p gg T , µ, δ q is the spherical Whittaker function W p g , 2 µ, ψ 2 I q , up to a function C p µ, δ q so for y P Y ` , K w l p y , µ, δ q “ C p µ, δ q p ρ { 2 p y q W p? y , 2 µ, ψ 2 I q from Hashizume’s solution of the Whittaker differential equations: ÿ C w p µ, δ, sgn p y qq J w l p y , µ w q , K w l p y , µ, δ q “ w P W ÿ a m p µ qp 4 π 2 y q m J w l p y , µ q “ p ρ ` µ p y q m P N n ´ 1 0 ˜ n ´ 1 ¸ n ´ 1 n ´ 1 ÿ ÿ ÿ p m j ´ m j ` 1 q 2 ´ m n ´ j p µ j ` 1 ´ µ j q a m p µ q “ a m ´ e j p µ q j “ 0 j “ 1 j “ 1 a 0 p µ q “ 1 , m 0 “ m n “ 0 . Jack Buttcane Bessel functions outside GL p 2 q . 8/14
What do we know about these Bessel functions? The uniqueness conjecture is true in the long element case: p ´ ρ p g q K w l p gg T , µ, δ q is the spherical Whittaker function W p g , 2 µ, ψ 2 I q , up to a function C p µ, δ q so for y P Y ` , K w l p y , µ, δ q “ C p µ, δ q p ρ { 2 p y q W p? y , 2 µ, ψ 2 I q from Hashizume’s solution of the Whittaker differential equations: ÿ C w p µ, δ, sgn p y qq J w l p y , µ w q , K w l p y , µ, δ q “ w P W ÿ a m p µ qp 4 π 2 y q m J w l p y , µ q “ p ρ ` µ p y q m P N n ´ 1 0 ˜ n ´ 1 ¸ n ´ 1 n ´ 1 ÿ ÿ ÿ p m j ´ m j ` 1 q 2 ´ m n ´ j p µ j ` 1 ´ µ j q a m p µ q “ a m ´ e j p µ q j “ 0 j “ 1 j “ 1 a 0 p µ q “ 1 , m 0 “ m n “ 0 . The asymptotics determine C w p µ, δ, sgn p y qq . Jack Buttcane Bessel functions outside GL p 2 q . 8/14
What do we know about these Bessel functions? What about the other Weyl elements? Jack Buttcane Bessel functions outside GL p 2 q . 9/14
What do we know about these Bessel functions? What about the other Weyl elements? The long element case of the interchange of integrals is the most difficult (highest dimension), the others are only difficult because there are many things to keep track of. Jack Buttcane Bessel functions outside GL p 2 q . 9/14
What do we know about these Bessel functions? What about the other Weyl elements? The long element case of the interchange of integrals is the most difficult (highest dimension), the others are only difficult because there are many things to keep track of. The positive sign case is no longer the spherical Whittaker function, so we lose Hashizume’s (and Harish-Chandra’s) and Stade’s work, so we have to work out the differential equations and power series solutions ourselves. Jack Buttcane Bessel functions outside GL p 2 q . 9/14
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