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Weyl Group Multiple Dirichlet Series

Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory (AM-175)

Ben Brubaker
Daniel Bump
Solomon Friedberg
Copyright Date: 2011
Pages: 170
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  • Book Info
    Weyl Group Multiple Dirichlet Series
    Book Description:

    Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics.

    These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished.

    The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation.

    eISBN: 978-1-4008-3899-8
    Subjects: Mathematics

Table of Contents

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  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-vi)
  3. Preface
    (pp. vii-xiv)
  4. Chapter One Type A Weyl Group Multiple Dirichlet Series
    (pp. 1-9)

    We begin by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, we choose the following parameters.

    Φ, a reduced root system. Let$ r $denote the rank of Φ.

    $ n $, a positive integer

    $ F $, an algebraic number field containing the group$ \mu_ {2n} $of$ 2n $-th roots of unity,

    $ S $, a finite set of places of$ F $containing all the archimedean places, all places ramified over$ \mathbb{Q} $, and large enough so that the ring

    \[{{\mathfrak{o}}_{S}}=\{x\in F|\ | x \right|_{\upsilon }\leqslant 1\text{ for }\upsilon \notin S\}\]

    of$ S $-integers is a principal ideal domain,$ m = (m _1 , \cdots ,m_r )$, an$ r $-tuple of nonzero...

  5. Chapter Two Crystals and Gelfand-Tsetlin Patterns
    (pp. 10-21)

    We will translate the definitions of the Γ and Δ arrays in (1.11), and hence of the multiple Dirichlet series, into the language of crystal bases. The entries in these arrays and the accompanying boxing and circling rules will be reinterpreted in terms of the Kashiwara operators. Thus, what appeared as a pair of unmotivated functions on Gelfand-Tsetlin patterns in the previous chapter now takes on intrinsic representation theoretic meaning. Despite the conceptual importance of this reformulation, the reader can skip this chapter and the subsequent chapters devoted to crystals with no loss of continuity. For further background information on...

  6. Chapter Three Duality
    (pp. 22-25)

    The material contained in this chapter won’t be used again until Chapter 18 and so can be skipped without loss of continuity. Its purpose is to point out that the boxing and circling decorations of the BZL patterns that were introduced in the last chapter are in a sense dual to each other.

    In Chapter 2 we used either of the notations

    \[\upsilon \;\left[ {\begin{array}{*{20}{c}} {{b_1}} & \cdots & {{b_N}} \\ {{i_1}} & \cdots & {{i_N}} \\ \end{array} } \right]\upsilon '\quad {\text{or}}\quad \upsilon \;{\left[ {\begin{array}{*{20}{c}} {{b_1}} & \cdots & {{b_N}} \\ {{i_1}} & \cdots & {{i_N}} \\ \end{array} } \right]^{(f)}}\upsilon '\]

    to mean that${\upsilon }'=f_{{{i}_{N}}}^{{{b}_{N}}}\cdots f_{{{i}_{1}}}^{{{b}_{1}}}\,\upsilon$where each integer$b_{k}$is as large as possible in the sense that$f_{{{i}_{k}}}^{{{b}_{k}}+1}\,f_{{{i}_{k-1}}}^{{{b}_{k-1}}}\cdots f_{{{i}_{1}}}^{{{b}_{1}}}\,\upsilon =0$. In this chapter, we will exclusively use the second notation – the superscript ($ f $) will be needed...

  7. Chapter Four Whittaker Functions
    (pp. 26-30)

    Weyl group multiple Dirichlet series are expected to be Whittaker coefficients of metaplectic Eisenstein series. The fact that Whittaker coefficients of Eisenstein series reduce to the crystal description that we gave in Chapter 2 is proved for Type A. In a classical setting, this was established by Brubaker, Bump, and Friedberg [13]. Alternatively, on the adele group, the corresponding local computation reduces to the evaluation of a type of$p$-adic integral. These were considered by McNamara [65], who reduced the integrals to sums over crystals by a very interesting method. A full treatment of this topic is outside the...

  8. Chapter Five Tokuyama’s Theorem
    (pp. 31-35)

    Let$z=\text{diag}({{z}_{1}},\cdots ,{{z}_{r+1}})$be an element of the group$\hat{T}(\mathbb{C})$, which is the diagonal subgroup of$\mathrm{GL}_{r+1}(\mathbb{C})$. In the application to the Casselman-Shalika formula we will take the$z_{i}$to be the Langlands parameters. (In terms of the$s_{i}$the$z_{i}$are determined by the conditions that$\prod{{{z}_{i}}=1}$and${{{z}_{i}}}/{{{z}_{i+1}}}\;=\ {{q}^{1-2{{s}_{r+1-i}}}}$.)

    Let us write the Weyl character formula in the form

    \[\left[ {\prod\limits_{\alpha \in \Phi ^ + } {(1\; - \;z^\alpha )} } \right]\;s_\lambda (z) = z^\rho \sum\limits_{w \in W} {( - 1)^{l(w_0 w)} z} ^{w(\lambda + \rho )}, \caption{(5.1)} \]

    where we recall that${{s}_{\lambda }}(z)={{s}_{\lambda}}({{z}_{1}},\cdots ,{{z}_{r+1}})$is the Schur polynomial. The sum over the Weyl group on the right-hand side is thenumeratorin the Weyl formula and the product on the left is essentially theWeyl denominator....

  9. Chapter Six Outline of the Proof
    (pp. 36-50)

    The proof of Theorem 1.1 involves many remarkable phenomena, and we wish to explain its structure in this chapter. To this end, we will give the first of a succession of statements, each of which implies the theorem. Passing from each statement to the next is a nontrivial reduction that changes the nature of the problem to be solved. We will outline the ideas of these reductions here and tackle them in detail in subsequent chapters.

    Statement A.We have$H_{\Gamma} = H_{\Delta}$.

    This reduction was already mentioned in the first chapter, where Statement A appeared as Theorem 1.2.

    The proof that...

  10. Chapter Seven Statement B Implies Statement A
    (pp. 51-53)

    In this chapter we will recall the use of the Schützenberger involution on Gelfand-Tsetlin patterns in [15] to prove that Statement B implies Statement A. We will return to these statements two more times in the later chapters of the book. In Chapter 18 we will reinterpret both Statements A and B in terms of crystals, and directly prove that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 18.2. Then in Chapter 19 we will yet again reinterpret Statements A and B in a different context, and yet again directly prove that the reinterpreted Statement B implies...

  11. Chapter Eight Cartoons
    (pp. 54-57)

    This chapter will introduce a method of marking up a short Gelfand-Tsetlin pattern based on inequalities between its entries, that encodes the effect of the involution$\mathfrak{t} \mapsto \mathfrak{t}^{\prime}$and the boxing and circling of its accordion. This will have another benefit: it will lead to the decomposition of the pattern into pieces calledepisodesthat will ultimately lead to the reduction to the totally resonant case.

    Proposition 8.1(i) If$n \nmid a$then$h(a) = 0$and$\left| g(a) \right| ={{q}^{a-\frac{1}{2}}}$.

    (ii) If$n|a$then

    \[h(a) = \phi (p^a ) = q^{a - 1} (q - 1),\quad g(a) = - q^{a - 1} \]

    (iii) If$n|a$and$b > 0$then

    \[h(a + b) = q^a h(b)\:,\quad g(a + b) = q^a g(b).\]

    (iv) If$n \nmid a, b$but$n|a + b$then

    \[g(a)g(b) = q^{a+b-1}.\]

    Properly (iii) means that$g^{\flat}$and$h^{\flat}$are periodic with period$n$.


  12. Chapter Nine Snakes
    (pp. 58-63)

    The key lemma of this chapter was stated without proof in [15], where it was called the “Snake Lemma.” (It is of course not the famous Snake Lemma from homological algebra.) Here we will recall it, prove it, and use it to prove the statement made in Chapter 6, that (6.13) is “often” true.

    By anindexingof the Γ preaccordion we mean a bijection

    \[\phi :\{1,\ 2,\cdots ,\ 2d+1\}\ \to \ {{\Theta }_{B}}\].

    With such an indexing in hand, we will denote$ \Gamma _{\mathfrak{t}}(\alpha) $by$\gamma _{k}(\mathfrak{t})$or just$\gamma _{k}$if$\alpha = \phi (k) $corresponds to$k$. Thus

    \[\{ \gamma _1 ,\gamma _2 , \cdots ,\gamma _{2d + 1} \} = \{ \Gamma (\alpha )|a \in \Theta _B \} \].

    We will also consider an indexing ψ of the$\Delta ^{\prime}$...

  13. Chapter Ten Noncritical Resonances
    (pp. 64-66)

    We recall that a short pattern (6.2) isresonantat$ i $if$ l_{i+1}=b_{i} $. This property depends only on the associated prototype, so resonance is actually a property of prototypes. We also call a first (middle) row entry$ a_{i} $criticalif it is equal to one of its fourneighbors, which are$ l_{i}, l_{i+1},b_{i} $and$ b_{i-1} $. We say that the resonance at$ i $iscriticalif either$ a_{i} $or$ a_{i+1} $is critical.

    Theorem 10.1Suppose that$ \mathfrak{t} $is a strict pattern with no critical resonances; then$ \mathfrak{t}' $is also strict with no critical resonances. Choose canonical indexings$ \gamma _{i} $and$ \delta_{i}^{\prime} $as in Proposition 9.1...

  14. Chapter Eleven Types
    (pp. 67-73)

    We now divide the prototypes into much smaller units that we calltypes. We fix a top and bottom row, and therefore a cartoon. For each episode$ \mathcal{E} $of the cartoon, we fix an integer$k _{\mathcal{E}} $. Then the set$ \mathfrak{S} $of all short Gelfand-Tsetlin patterns (6.2) with the given top and bottom rows such that for each$ \mathcal{E} $

    \[\sum\limits_{\alpha \in \Theta _1 \cap \mathcal{E} }{\frak{t}(\alpha )} = k_\mathcal{E} \caption{(11.1)}\]

    is called atype. Thus two patterns are in the same type if and only if they have the same top and bottom rows (and hence the same cartoon), and if the sum of the first (middle) row elements...

  15. Chapter Twelve Knowability
    (pp. 74-76)

    We turn now to the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in$ q $, the order of the residue class field. We refer to Chapter 6 for additional discussion of knowability and its role in the proof of Theorem 1.2.

    Let$\mathfrak{S}={{\prod{\mathfrak{S}}}_{i}}$be a type. Let$ \mathcal{E}=\mathcal{E}_{i} $be an episode in the cartoon associated to the short Gelfand-Tsetlin pattern$ \mathfrak{t}\in \mathfrak{S} $. If the episode is of Class II, let${{a}_{0}},\cdots ,{{a}_{d}}$and${{L}_{1}},\cdots ,{{L}_{d+1}}$be as in Proposition 11.4. If the class is I, III, or IV, we...

  16. Chapter Thirteen The Reduction to Statement D
    (pp. 77-86)

    We now switch to the language of resotopes, as defined in Chapter 6. We remind the reader that we may assume$ \gamma _{L _{\varepsilon}} $and$ \gamma _{R _{\varepsilon}} $are multiples of$ n $for every totally resonant episode. We also recall that$ s $and$ d $are the weights of the accordions (6.15) and (6.17) under consideration.

    Proposition 13.1Statement D is equivalent to Statement C. Moreover, Statement D is true if$ n \nmid s $.

    Proof. The case of a totally resonant short Gelfand-Tsetlin pattern$ \mathfrak{t} $is a special case of Proposition 11.4, and the point is that$ \Gamma _{\mathfrak{t}} $is a Γ-accordion$ \mathfrak{a} $, and Proposition 11.4 shows...

  17. Chapter Fourteen Statement E Implies Statement D
    (pp. 87-88)

    We fix a nodal signature η. Let$B(\eta )=\left\{ i|{{\eta }_{i}}=\square \right\}$. Let\[\mathcal{C}{\mathcal{P}_\eta }({c_0}, \cdots ,{c_d}) \in {\mathfrak{Z}_\Gamma }$be the following “cut and paste” virtual resotope\[\mathcal{C}{\mathcal{P}_\eta }({c_0}, \cdots ,{c_d}) = \sum\limits_{T \subseteq B(\eta )} {{{( - 1)}^{|T|}}{\mathcal{A}_s}(c_0^T,} \cdots ,c_d^T), \caption{(14.1)}\]where\[c_i^T = \left\{ {\begin{array}{*{20}{c}} {{c_i}\quad {\text{if}}\ i \in T,} \\ {\infty \quad {\text{if}}\ i \notin T.} \\ \end{array} }\right.\]We recall that the simplex$\text{C}{{\text{P}}_{\eta }}({{c}_{0}},\cdots ,{{c}_{d}})$is the set of Γ-accordions\[\mathfrak{a} = \left\{ {\begin{array}{*{20}{c}} s & {} & {{\mu _1}} & {} & \cdots & {} & {{\mu _d}} \\ {} & {{\nu _1}} & {} & \cdots & {} & {{\nu _d}} & {} \\ \end{array} } \right\}\]

    that satisfy the inequalities (6.22), with the convention that$ \mu _{0} = s $and$ \mu _{d+1} = 0 $. Geometrically, this set is a simplex, and we will show that it is the support of$\mathcal{C}{\mathcal{P}_\eta }({c_0}, \cdots ,{c_d})$, though the latter virtual resotope is a superposition of resotopes whose supports include elements that are outside of$\mathrm{CP}}_{\eta }}({{c}_{0}},\cdots ,{{c}_{d}})$; it will be shown that the alternating sum causes such terms to cancel.

    Finally, if$\mathfrak{a} \in \mathrm{CP}_{\eta}(c_{0}, \cdots, c_{d})$let...

  18. Chapter Fifteen Evaluation of $ \Lambda _{\Gamma} $ and $ \Lambda _{\Delta} $, and Statement G
    (pp. 89-95)

    Let η be a nodal signature, and let σ be a subsignature. Let\[\mathfrak{a} = \left\{ {\begin{array}{*{20}{c}} s & {} & {{\alpha _1}} & {} & {{\alpha _2}} & \cdots & {} & {{\alpha _d}} \\ {} & {{\beta _1}} & {} & {{\beta _2}} & {} & \cdots & {{\beta _d}} & {} \\ \end{array} } \right\}\]be an accordion belonging to the open facet$ \mathcal{S} _{\sigma} $of$\text{C}{{\text{P}}_{\eta }}({{c}_{0}},\cdots ,{{c}_{d}})$. Assuming that$ n|s $we will evaluate$ \Lambda _{ \Gamma } (\mathfrak{a}, \sigma )$.

    We will denote\[V(a,b)={{(q-1)}^{a}}{{q}^{(d+1)s-b}},\quad V(a)=V(a,a).\]Let\[ {{\varepsilon _\Gamma }(\sigma ) = {\varepsilon _\Gamma } = \left\{ {\begin{array}{*{20}{c}} {1\quad {\text{if}}\ {\sigma _0} = \square ,} \\ {0\quad {\text{otherwise,}}} \\ \end{array} } \right.} \]\[ \begin{array}{*{20}{c}} {{\mathcal{K}_\Gamma }(\sigma ) = {\mathcal{K}_\Gamma } = \left\{ {i|1 \leqslant i \leqslant d,{\sigma _i} = \square ,{\sigma _{i - 1}} \ne \circ} \right\},\quad {k_\Gamma } = \left| {{\mathcal{K}_\Gamma }} \right|,} \\ {{\mathcal{N}_\Gamma }(\sigma ) = {\mathcal{N}_\Gamma } = \left\{ {i|1 \leqslant i \leqslant d,{\sigma _i} = \square ,{\sigma _{i - 1}} = \circ} \right\},\quad {n_\Gamma } = \left| {{\mathcal{N}_\Gamma }} \right|,} \\ \end{array} \]and\[\begin{array}{*{20}{r}} {{\mathcal{C}_\Gamma }(\sigma ) = {\mathcal{C}_\Gamma } = } \\ {\left\{ {i|1 \leqslant i \leqslant d,{\sigma _0},{\sigma _1}, \cdots ,{\sigma _{i - 1}}\;{\text{not all}}\; \circ\ {\text{and either }}i \in {\mathcal{N}_\Gamma }\;{\text{or}}\;{\sigma _i} = \ast} \right\}}. \\ \end{array} \caption{(15.1)} \]

    Let$ c_{\Gamma}=|\mathcal{C}_{\Gamma}| $, and let$ t_{\Gamma} $be the number of$ i $with$ 1 \leqslant i \leqslant d $and$ \sigma _{i} \neq * $. Given a set of indices$\Sigma{=\left\{ {{i}_{1}},\cdots ,{{i}_{k}} \right\}}\subseteq \left\{ 1,2,\cdots ,d \right\}$, let\[{\delta _n}({i_1}, \cdots ,{i_k}) = {\delta _n}(\Sigma ) = {\delta _n}(\Sigma ;\mathfrak{a}) = \left\{ \begin{array}{ll} 1 & {\text{if}}\ n\ {\text{divides}}\ {\alpha _{{i_1}}}, \cdots ,{\alpha _{{i_k}}}, \\ 0 & {\text{otherwise}}. \\ \end{array} \right. \caption{(15.2)}\]Let\[\chi _\Gamma (\mathfrak{a},\sigma ) = \chi _\Gamma = \prod\limits_{i \in\mathcal{C}_\Gamma (\sigma )} {\delta _n (i)} \]. Finally, let\[{a_\Gamma }(\sigma ) = {a_\Gamma } = 2(d - {t_\Gamma } + {n_\Gamma }) + \left\{ \begin{array}{cl} { - 1\quad \mathrm{if}\ {\sigma _0} = \circ } \\ {0\quad \mathrm{if}\ {\sigma _0} = \square } \\ {1\quad \mathrm{if}\ {\sigma _0} = *} \\ \end{array} \right\} + \left\{ {\begin{array}{*{20}{c}} {1\quad \mathrm{if}\ {\sigma _d} = \circ } \\ {0\quad \mathrm{if}\ {\sigma _d} \ne \circ } \\ \end{array} } \right\}.\]

    Proposition 15.1Assume that$ n|s $.Given aΓ-accordion\[\mathfrak{a} = \left\{ {\begin{array}{*{20}{c}} s & {} & {{\alpha _1}} & {} & {{\alpha _2}} & \cdots & {} & {{\alpha _d}} \\ {} & {{\beta _1}} & {} & {{\beta _2}} & {} & \cdots & {{\beta _d}} & {} \\ \end{array} } \right\}\]

    and an associated signature$\sigma \subseteq {{\sigma }_{\mathfrak{T}}}$not containing the sequence$ \circ, \square $,then

    \[{\mathcal{G}_{\Gamma}}(\mathfrak{a},\sigma ) = {( - 1)^{{\varepsilon _\Gamma }}}{\chi _\Gamma } \cdot V({a_\Gamma },{a_\Gamma } + {d_\Gamma }), \caption{(15.3)}\]where\[ d_{\Gamma} = \left(\sum_{\begin{smallmatrix}1 \leqslant i \leqslant d\\ \sigma_{i}=\square\end{smallmatrix}} (1 + \delta_{n}(i))\right) + \left\{ \begin{array}{ll} 1 & if \ \sigma_{0} = \square\\ 0 & if \ \sigma_{0} \neq \square \end{array}\right\}. \]

    Recall that any subsignature$\sigma$containing the string$\circ\ \square$has...

  19. Chapter Sixteen Concurrence
    (pp. 96-103)

    This chapter contains purely combinatorial results that are needed for the proof of Statement G. The motivation for these results comes from the appearance of divisibility conditions through the factor$ \delta _{n}(\Sigma ;\mathfrak{a}) $defined in (15.2) that appears in Theorems 15.3 and 15.4. We refer to the discussion of Statement G in Chapter 6 for an informal discussion of the results of this chapter.

    Let$ 0\leqslant f\leqslant d $. In Chapter 6 we defined bijections$ {\phi }_{\sigma ,\tau }}:{\mathcal{S}_{\sigma }}\to {\mathcal{S}_{\tau }$between the open$ f $-facets, and a related equivalence relation, whose classes we call$ f $-packets. According to Statement F, the sum of$ \Lambda_{\Gamma}(\mathfrak{a}, \sigma ) $over an$ f $-packet...

  20. Chapter Seventeen Conclusion of the Proof
    (pp. 104-107)

    In Chapter 15 we reduced the proof of Theorem 1.2 to Statement G, given at the end of that chapter, and we now have the tools to prove it.

    Lemma 17.1The cardinality of eachΓ-pack orΔ-pack is a power of 2.

    Proof. In a Γ-swap$ \ast \square $is replaced by$ \circ \ast $in the signature. Since both signatures are subsignatures of η, this means that η has$ \circ \square $at this location. From this it is clear that if a Γ-swap is possible at$ i-1, i $then no swap is possible at$ i-2, i-1 $, or$ i, i+1 $, and so the swaps are...

  21. Chapter Eighteen Statement B and Crystal Graphs
    (pp. 108-114)

    Theorem 1.2 is now proved, but we have further important remarks to make, which will occupy the last three chapters. These chapters may be read independently of each other.

    In this chapter, we will translate Statements A and B from Chapter 6 into Statements A′ and B′ in the language of crystal bases, and explain in this language how Statement B′ implies Statement A′.

    We slightly generalize the definition (2.14) by considering more general reduced words Ω and defining

    \[G_\Omega (\upsilon ) = \prod\limits_{b_i \in {\text{BZL}}_\Omega (\upsilon )} {\left\{ {\begin{array}{ll} {g(b_i )} & {{\text{if }}b_i {\text{ is boxed but not circled in BZL}}_\Omega (v),} \\ {q^{b_i } } & {{\text{if }}b_i {\text{ is circled but not boxed,}}} \\ {h(b_i )} & {{\text{if }}b_i {\text{ is neither circled nor boxed,}}} \\ 0 & {{\text{if }}b_i {\text{ is both boxed and circled}}.} \\ \end{array} } \right.} \]

    This definition is provisional since it assumes that we can give an appropriate definition of boxing and circling for Ω....

  22. Chapter Nineteen Statement B and the Yang-Baxter Equation
    (pp. 115-131)

    The$ p $-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was introduced by Baxter [1] for proving the commutativity ofrow transfer matricesfor the six-vertex and similar models. This is significant for us, because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This last reformulation opens...

  23. Chapter Twenty Crystals and $ p $-adic Integration
    (pp. 132-142)

    In many cases, integrations of representation theoretic import over the maximal unipotent subgroup of a$ p $-adic group can be replaced by a sum over Kashiwara’s crystal$ \mathcal{B}(\infty) $. Partly motivated by the crystal description presented in Chapter 2 of this book, this perspective was advocated by Bump and Nakasuji in [21]. Later work by McNamara [65] and Kim and Lee [50] extended this philosophy yet further. Indeed, McNamara shows that the computation of the metaplectic Whittaker function is initially given as a sum over$ \mathcal{B}(\infty) $, which, upon closer examination, degenerates to the sum over$ \mathcal{B}_{\lambda + \rho } $presented in Chapter...

  24. Bibliography
    (pp. 143-148)
  25. Notation
    (pp. 149-154)
  26. Index
    (pp. 155-158)