Before coming to this, let us quickly discuss global hypergeometric functions , which seem the main application of DAHA with known and expected applications well beyond harmonic analysis. In the $ p$-adic limit, which is $ q\to 0$, they reduce to the polynomials; classical hypergeometric functions are for $ q\to 1$. Global functions $ \Phi_{q,t}(X,\mmLa) ,\ q< 1$. ([Cherednik1997]) These functions are defined as reproducing kernels of the DAHA Fourier Transform; no difference equations are used in this approach for their definition. To calculate them we use that this transform sends Laurent polynomials in terms of $ X=q^x$ times the Gaussian $ q^{-x^2/2}$ to such polynomials times $ q^{+x^2/2}$, which is similar to theory of Hankel transform. This gives an explicit formula for the series $ {\tilde{\Phi}_{q,t}(X,\mmLa)\!\equal\! \theta_R(X)\theta_R(\mmLa)\Phi_{q,t}(X,\mmLa)}$ in terms of $ P_\mu(X)P_\mu(\mmLa)$ for Macdonald polynomials $ P_\mu$ (see below for $ A_1$). This is general theory, for any (reduced, irreducible) root systems $ R$. The Laurent polynomials are in terms of $ X_\la=q^{(x,\la)}$ for $ \la\in P$ (the weight lattice for $ R$), $ (\cdot,\cdot)$ is the standard $ W$-invariant inner product, $ x^2=(x,x)$, $ \theta_R$ is the usual theta-series associated with $ R$, $ \mu\in P$. This is from [Cherednik1997]; see also [Cherednik2006], [Cherednik2009b].

The series $ \tilde{\Phi}_{q,t}(X,\mmLa)$ is absolutely convergent as $ |q|< 1$, $ W$-invariant with respect to both, $ X$ and $ \mmLa$, and, importantly, $ X\!\leftrightarrow\! \mmLa$-symmetric (as for the Bessel functions). Here one must avoid the poles of the coefficients of Macdonald polynomials and zeros of $ \theta_R(X)$; otherwise the convergence is really global. Such global functions are missing in the (differential) Harish-Chandra theory.

Furthermore, let $ X=q^x,\mmLa=q^\la$. Assume that $ \la=w(\la_+)$ for dominant $ \la_+$ such that $ \Re(\la_+,\al_i)> 0$, i.e. that $ \la$ is generic. Then $ \Phi_{q,t}(X,\mmLa)$ under some explicit normalization becomes an asymptotic series $ \Phi^{a\!s}_{q,t}(X,\mmLa)= q^{-(x,\la_+)} t^{(x+\la_+,\rho)}(1+\ldots)$ as $ \Re(x,\al_i)\to +\infty$. Harish-Chandra decomposition. See [Cherednik2009b], [Cherednik and Orr2011], [Stokman2014]. It is: \begin{eqnarray*} \Phi_{q,t}(X,\mmLa)= \sum_{w\in W} \si_{q,t}(w(\mmLa))\, \Phi_{q,t}^{a\!s}(X,w(\mmLa))\ \end{eqnarray*} for the $ q,t$-extension $ \si_{q,t}(\mmLa)$ of the Harish-Chandra $ c$-function. Note that (naturally) $ \l_p\Phi_{q,t}(X,\mmLa)=p(\mmLa)\Phi_{q,t}(X,\mmLa)$ for $ p\in \C[X]^W$ and Macdonald–Ruijsenaars operators $ \l_p$ in type $ A$ (and due to DAHA for any root systems). However this relation is used only a little in the formula/theory of $ \Phi_{q,t}$. The recovery formula is important here; $ P_\lambda$ proportional to $ \Phi_{q,t}(X,\mmLa)$ for $ \Lambda=t^{\rho}q^{\lambda}$ (with an explicit coefficient of proportionality).

Let us give the exact formulas for $ A_1$. For Rogers-Macdonald polynomials $ P_n(X)$, $ \mu$ provided below, and $ \theta(X)\equal\sum_{m=-\infty}^\infty q^{mx+m^2/4}$, \begin{align*} &\frac{\theta(X)\theta(\mmLa)}{\theta(t^{1/2})}\Phi= \tilde{\Phi}_{q,t}(X,\mmLa)\equal\, \sum_{n=0}^\infty\,q^{\frac{n^2}{4}}\, t^{\frac{n}{2}} \,\frac{P_n(X)P_n(\mmLa)\, (\mu)_{\mathrm{CT}}} {(P_n P_n\,\mu)_{\mathrm{CT}}},\ |q|< 1. \end{align*}

For $ |X|\!< \!|t|^{\frac{1}{2}}|q|^{-\frac{1}{2}}$, the Harish-Chandra formula reads: $ \tilde{\Phi}_{q,t}(X,\mmLa)\!=$ \begin{align*} =&\,(\mu)_{\mathrm{CT}}\,\si(\mmLa)\,\,\theta(X\mmLa t^{-1/2}) \,\sum_{j=0}^\infty\, (\frac{q}{t})^{\!{}^{j}}\,X^{2j}\prod_{s=1}^j\frac{(1-tq^{s-1}) (1-q^{s-1}t\mmLa^{-2})}{(1-q^s)(1-q^s\mmLa^{-2})} \notag\\ +&\,(\mu)_{\mathrm{CT}}\, \si(\mmLa^{-1})\theta(X\mmLa^{-1}t^{-1/2})\, \sum_{j=0}^\infty (\frac{q}{t})^{\!{}^{j}}X^{2j}\prod_{s=1}^j\frac{(1\!-\!tq^{s-1}) (1\!-\!q^{s\!-\!1}t\mmLa^{2})}{(1\!-\!q^s)(1\!-\!q^s\mmLa^{2})}, \notag \end{align*} where $ \si(\mmLa)=\prod_{j=0}^\infty \frac{1-tq^j \mmLa^2}{1-q^j \mmLa^2}$ is the $ q,t$-generalization of the Harish-Chandra $ c$-function; $ (\cdot)_{\mathrm{CT}}$ is the constant term.

See [Cherednik and Orr2011]. The sums here are nothing but (special) basic Haine's hypergeometric functions. Letting here $ t\to 0$ in type $ A_n$, the asymptotic expansions of the resulting global $ q$-Whittaker function are essentially the Givental–Lee functions. This is an important connection between the physics $ B$-model (the usage of the global function) and the $ A$-model (the usage of its asymptotic expansions). We note that $ \Phi$ is actually an entirely algebraic object, uniquely determined by its asymptotic behavior, including the walls (resonances), when $ \Re(\al,\la)=0$ for some roots $ \al$ (the theory of resonances is still incomplete).

Let $ R\subset {\R}^n$ be a root system, $ Q\subset P$ (the weight lattice), $ W=\lan\!s_\al\!\!\ran$ for $ \al\in R$, $ \tilde{W}=W\lsmash Q\subset \hat{W}= W\lsmash P=\tilde{W}\lsmash \Pi$, where $ \Pi\!=\!P/Q$.

Then $ {\h\equal \lan\Pi,\, T_i(0\le i\le n)\ran /}$ $ \{$homogeneous Coxeter relations for $ T_i$, and $ (T_i-t^{\frac{1}{2}})(T_i+t^{-\frac{1}{2}})=0\for 1\le i\le n$ $ \}$, where $ {\R}$ will be the ring of coefficients, including $ q,t^{\pm 1/2}$. This is convenient to avoid the complex conjugation in the scalar products (and for positivity).

We set $ T_{\hw}\!=\!\pi T_{i_l}\!\ldots\! T_{i_1}$ for reduced decompositions $ \hw\!=\!\pi s_{i_l}\!\ldots\! s_{i_1}\!\in\! \hat{W},$ where $ l\!=\!l(\hw)$ is the length of $ \hw$. The canonical anti-involution, trace and scalar product are: \begin{align*} T_{\hw}^\star\equal T_{\hw^{-1}},\, \lan T_{\hw}\ran =\de_{id,\hw}, \quad \lan f,g\ran \equal \lan f^\star g\ran= \sum_{\hw\in \hat{W}}\,c_{\hw}d_{\hw}, \end{align*} where $ f\!=\!\sum c_{\hw} T_{\hw},\ g\!=\! \sum d_{\hw} T_{\hw}\, \in\, L^2(\h)=\{f,\, c_{\hw}\in {\R},\ \sum c_{\hw}^2< \infty \}$.

According to Dixmier, $ \lan f,g\ran= \int_{\pi\in\h^\vee} \hbox{Tr}(\pi(f^\star g))d\nu(\pi)$. We omit here some analytic details concerning the classes of functions. In the spherical case (referred to as "sph" later on), one takes $ f,g\in P_+\h P_+$, where $ {P_+\equal\sum_{w\in W} t^{\frac{l(w)}{2}}T_{w}.}$ The measure reduces correspondingly.

Macdonald found an integral formula for $ \nu_{sph}(\pi),$ as $ t> 1$. Its extension to $ 0< t< 1$ (due to ... Arthur, Heckman–Opdam, ...) by the analyticity is sometimes called "picking up residues" ([Ciubotaru et al.2012],[Heckman and Opdam1996], [Opdam2006], [Opdam and Solleveld2010]). The final formula (for any $ t$) generally reads: \begin{eqnarray*} \int \{\cdot\}\, d\nu_{sph}^{an}(\pi)= \sum C_{s,S}\cdot\int_{s+iS}\{\cdot\} \,d\nu_{s,S}, \end{eqnarray*} summed over (affine) residual subtori $ s+S.$ Residual points (very interesting and the most difficult to reach) correspond to square integrable irreducible modules (as their characters $ \chi_\pi$ extend to $ L^2(\h)$).

This formula involves deep algebraic geometry, the Kazhdan–Lusztig theory ([Kazhdan and Lusztig1987], [Lusztig1990]). In our approach via DAHA, this very formula expected to be a reduction of the analytic continuation of the DAHA inner product in the integral form, which requires only $ q$-calculus. The main claim is as follows. It is in the spherical case and is a theorem for any (reduced, irreducible) root systems, with an important reservation that the explicit formula is known by now only in type $ A$ (unpublished). The $ q,t$-generalization of the picking up residues is the presentation of the inner product in the DAHA polynomial representation as sum of integrals over DAHA residual subtori. Only the whole sum satisfies the DAHA invariance, and the corresponding $ C$-coefficients are uniquely determined by this property. Upon the limit $ q\to 0$, this approach potentially provides explicit formulas for the $ C_{s,C}$-coefficients above, including formal degrees (for the residual points).

We will now switch to the DAHA harmonic analysis. In contrast to the Harish-Chandra theory, where we mainly have two theories based on the imaginary and real integration, the so-called compact and non-compact cases, here we have more options. Let us try to outline them, disregarding various (many) specializations and the open project aimed at the passage from the $ q$-Gamma function in DAHA theory to the $ p$-adic Gamma (this is doable, but there are no works on this so far).

We think that there are essentially $ 6$ major theories by now, corresponding to different choices of "integrations"; some connections are shown by arrows. We stick to the imaginary integration in this paper.

As above, $ R\subset {\R}^n$ is a root system (irreducible and reduced), $ W$ denotes the Weyl group $ < s_i, 1 \leq i \leq n> ,\,\, P$ is the weight lattice.

We omit the general definition of DAHA (it will be provided later for $ A_1$); see [Cherednik2006]. The following will be sufficient. For $ T_w$ as above, \begin{eqnarray*} {\HH\!=\!\lan X_b,T_w,Y_b,q,t\ran,\, b\in P,w\in W,\ \, {\R}\ni t^{\pm1/2},\, q\!=\!\exp\left(-\frac{1}{a}\right), a> 0,} \end{eqnarray*} where the ring of coefficients is $ {\R}$. More formally, it is defined over $ \Z[q^{\pm\frac{1}{m}},t^{\pm\frac{1}{m}}]$ for proper $ m$.

We will stick to the polynomial case through this paper. Namely, $ \varrho$ will be the one-dimensional character of affine Hecke algebra $ \h_{Y}$ generated by $ T_w$ and $ Y_b$, which sends $ T_i\! \mapsto\! t^{1/2}, Y_b\!\mapsto\! t^{(\rho,b)}$ for $ i\ge 0, b\in P$. Here $ \rho=\frac{1}{2}\sum_{\al> 0}\al$. Generally, the number of different parameters $ t$ here equals the number of different lengths $ |\al|$ in $ R$. Then $ \{A,B\}$ acts via $ \x\!\times\! \x$ for the polynomial representation $ {\x={\R}[X^{\pm 1}]=Ind_{\h_Y}^{\HH} (\varrho)}$.

Generalizing the above definition, level-one anti-involutions $ \varkappa$ are such that dim$ {\HH/(\j\!+\!\j^\varkappa)\!=\!1}$ for $ {\x\!=\!\HH/\j}$, $ \j\!=\!\{H\mid H(1)=0\}$, $ 1\in \x$. The Shapovalov ones are obviously level-one. Then $ \{H\}_{\varkappa}^\varrho$ is defined as the image of $ H$ in $ {\HH/(\j\!+\!\j^\varkappa)}$. An example. Let $ \ast: g\mapsto g^{-1}$ for $ g=X_a,Y_b,T_w,q,t$. It is level-one for generic $ q,t$, but obviously not a Shapovalov anti-involution with respect to $ Y$. One can prove ([Cherednik2006]) that there exists the corresponding unique inner product in $ \x$ for generic $ q,t$ given generic $ q$, not for any $ t$.

For rational DAHA, the counterpart of $ \ast$ above (serving the "standard" inner product in $ \x$) is not level-one. The Rational DAHA is: \begin{eqnarray*} {\HH''\equal\langle x,y,s\rangle/\,\left\{\, [y,x]\!=\!\frac{1}{2}\!+\!k s,\ s^2\!=\!1,\ sxs\!=\!-x,\ sys\!=\!-y\,\right\}.} \end{eqnarray*} Accordingly the polynomial representation $ \x$ becomes $ {\R}[x]$ with the following action of $ {\HH''}$: $ \!s(x)=-x,\ \,x=$ multiplication by $ x,$ $ y\mapsto D/2,$ where $ D=\frac{d}{dx}+\frac{k}{x}(1-s)$ (the Dunkl operator).

Then the anti-involution $ x^*\!=\!x, y^*\!=\!-y, s^*\!=\!s$ formally serves the inner product $ \int f(x)g(x)|x|^{2k}$, but it diverges at $ \infty$. Algebraically, $ {\R}[x]$ has no $ *$-form for $ k\not\in -1/2-\Z_+$. Indeed, for $ p\in \Z_+$ $ \{1,y(x^{p})\}\!=\!0\!=\!\{1,c_{p}x^{p-1}\}$, where $ c_{2p}\!=\!p,\, c_{2p+1}$ $ =p+1/2+k$ (direct from the Dunkl operator). Hence, $ \{1,x^{p}\}=0$ ($ \forall p$) for non-singular $ k$ and $ \{\,,\,\}=0$.

To fix this problem, let us replace $ y$ by $ y+x$; then $ *$ becomes Shapovalov for such new $ y$ (the definition depends on the choice of $ y$). Indeed, the decomposition $ h=\sum c_{a\de b}((y+x)^*)^a s^\de (y+x)^b$ exists and is unique ($ \de=0,1$) for any $ h$. Defining the coinvariant by $ {\{h\}\equal\sum_{\de=0,1} c_{o\de o},\ \{f,g\}\equal\{f^*g\}}$, it acts through $ {\R}[x]e^{-x^2}\!\times {\R}[x]e^{-x^2}$ due to $ (y+x)e^{-x^2}=0$ for the natural action of $ {\HH''}$ on $ e^{-x^2}$. Indeed, $ {\R}[x]e^{-x^2}$ can be identified with $ {\HH''/(\HH''(y+x),\HH''(s-1))}$.

Explicitly, let $ p\!=\!\frac{a+b}{2}$ for $ a,b\in Z_+$. Then a direct PBW calculation readily gives that $ \, \{x^a ,x^b \}\!=\! (\frac{1}{2})^p(\frac{1}{2}+k)\cdots (\frac{1}{2}+k+p-1).$ Analytically, we ensured the convergence of $ \int_{{\R}}fg|x|^{2k}$ via the multiplication of $ f$ and $ g$ by $ e^{-x^2}$; let us provide the exact analysis. The integral presentation for this form is: \begin{eqnarray*} \{f\,,\,g\}= \frac{1}{i}\int_{-\ep+i{\R}}(fge^{-2x^2}\,(x^2)^k)dx/ (\cos(\pi k)C), \end{eqnarray*} where $ C=\Gamma(k+1/2)\,2^{k+1/2}, \ {\forall k\in \C},\ \ep> 0.$

For real $ k> -\frac{1}{2}\,$, one can simply do the following: \begin{eqnarray*} \{f\,,\,g\} =\frac{1}{i C}\int_{i{\R}} fge^{-2x^2}|x|^{2k}dx. \end{eqnarray*} Note using $ |x|$ here, which is not natural algebraically; one can take here $ x^{2k}$ instead using the technique of hyperspinors (see below).

Let $ k\!=\!-\frac{1}{2}\!-\!m\, (m\!\in\! \Z_+)$. Then we replace $ \int_{-\ep\!+i{\R}}\rightsquigarrow \frac{1}{2}(\int_{-\ep\!+i{\R}}\!+\! \int_{\ep\!+i{\R}})$ and $ \{f\,,\,g\}$ becomes $ \hbox{const Res}_0\,(fge^{-2x^2} x^{-2m-1}dx).$ The radical of this form is non-zero. It is $ (x^{2m+1}e^{-x^2})$, which is a unitary $ {\HH''}$-module with respect to the form $ \frac{1}{i}\int_{i{\R}} fge^{-2x^2}|x|^{-2m-1}dx$ restricted to this module (the convergence at $ x=0$ is granted). The $ *$-form of the quotient $ {\R}[x]/(x^{2m+1})$ is non-positive. See [Cherednik and Ma2013] for some details.

If $ t^{\frac{1}{2}}=1,$ then $ T^2=1$ and we will replace $ T$ by $ s$. In this case, $ {\HH}$ becomes the Thus DAHA unites Weyl algebras with the Hecke ones. The Heisenberg and Weyl algebra extended by $ \mathbf{S}_2\, $. I.e. the relations are: \begin{align*} &sXs\!=\!X^{-1}, \quad sYs\!=\!Y^{-1}, \quad Y^{-1}X^{-1}YX\!=\!q^{-1/2},\, s^2=1. \end{align*} Weyl algebras (also called non-commutative tori) are the main tools in quantization of symplectic varieties. So DAHA can be expected to serve "refined quantization" (with extra parameters) of varieties with global or local (in tangent spaces) $ W$-structures for Weyl groups $ W$.

The whole $ PSL_2(\mathbf{ Z})$ acts projectively in $ \b_q$ and $ {\HH}$: \begin{align*} &\binom{1\quad1}{ 0\quad1}\sim \tau_+: Y\mapsto q^{-1/4}XY,\ X\mapsto X,\ T\mapsto T,\\ &\binom{1\quad0}{ 1\quad1}\sim \tau_-: X\mapsto q^{1/4}YX,\,\ \, Y\mapsto Y,\ \, T\mapsto T. \end{align*}

They are directly from topology. The key for us is a pure algebraic fact that $ \tau_+$ is the conjugation by $ q^{x^2},$ where $ X=q^x$; use $ \x$ below to see this. DAHA FT is for $ \tau_+^{-1}\tau_-\tau_+^{-1}=\si^{-1}=$ $ \tau_-\tau_+^{-1}\tau_-.$

Generators of $ \b_1$ and relation $ Y^{\!-\!1}X^{\!-\!1}YXT^2 = 1$.

More exactly, the operator Fourier transform is the DAHA automorphism sending: $ q^{1/2}\mapsto q^{1/2}, t^{1/2}\mapsto t^{1/2}$, \begin{eqnarray*} Y\mapsto X^{-1},\ X\mapsto T Y^{-1}T^{-1},\ T\mapsto T; \end{eqnarray*} topologically, it is essentially the transposition of the periods of $ E$, though it is not an involution; it corresponds to the matrix $ \begin{pmatrix}0 & \quad-1\\ 1 &\quad0\end{pmatrix}$ representing $ \si^{-1}$. Polynomial representation. It was defined above as $ {Ind_{\h_Y}^{\HH}(\varrho).}$ It is in the space $ \x$ of Laurent polynomials of $ X=q^x.$ The action is: \begin{align*} &T\mapsto \,t^{1/2}s\,+\,\, \frac{t^{1/2}-t^{-1/2}} {q^{2x}-1}(s-1),\,\, Y\mapsto \pi T,\\ &{where\, \ } \pi=sp, sf(x)=f(-x), s(X)=X^{-1},\\ &pf(x)=f(x+1/2),\ p(X)=q^{1/2}X, \ t=q^k. \end{align*} Here $ Y$ becomes the difference Dunkl operator; $ X$ acts by the multiplication. The standard AHA stuff (Bernstein's Lemma) gives that $ Y+Y^{-1}$ preserves $ {\x_{sym}\equal}$ $ \{$symmetric (even) Laurent polynomials$ \}$; the difference operator $ Y+Y^{-1}\mid_{sym}$ is sometimes called the $ q,t$-radial part. Basic inner products. ([Cherednik2006]) Note that we do not conjugate $ q,t$ below (a simplest way to supply $ \x$ with an inner product).

For $ X\!=\!q^x,q\!=\!\exp(-\frac{1}{a})$ and the Macdonald truncated $ \theta$-function \begin{eqnarray*} \mu(x)=\prod_{i=0}^{\infty}\frac{(1-q^{i+2x})(1-q^{i+1-2x})} {(1-q^{i+k+2x})(1-q^{i+k+1-2x})},\, \hbox{ we set:} \end{eqnarray*} $ {\lan f,g \ran_{{1/4}}\!\equal\! \frac{1}{2\pi a i}\int_{1/4\!+\!P}\,f(x)\,T(g)(x) \mu(x) dx,}$ where $ P=[-\pi i a,\pi i a].$

Proof.

The ingredients are as follows: the Shapovalov $ \,\varkappa\,$ above (for $ Y$) and the standard coinvariant $ \varrho$ (serving $ \x$). Recall that \begin{eqnarray*} {\varrho\,\left(\sum_{a,b\in \Z}^{\ep=0,1}\, c_{a\ep b} (Y^\varkappa)^a\, T^\ep\, Y^b\right)\equal\sum c_{a\ep b} t^{\frac{a+\ep+b}{2}}} \end{eqnarray*} and the corresponding form is \begin{eqnarray*} {\{A,B\}_\varkappa^\varrho\,\equal\,\varrho(A^\varkappa B)= \{B,A\}_\varkappa^\varrho \text{ in } \HH\ni A,B.} \end{eqnarray*} The latter acts via $ \x\! \times\! \x$, $ \x\!=\!{\R}[X^{\pm 1}],$ and satisfies the normalization $ \{1,1\}\!=\!1$ by construction. This form is regular (analytic) for all $ k\in\C$.

Proof.

Importantly, $ \hat{\Phi}^k$ is meromorphic for $ \Re k\!> \!-1$ (i.e. beyond $ -\frac{1}{2}$ for $ \Phi_{\frac{1}{4}}^k$); so it coincides with $ G(k)\{f,g\}_\varkappa^\varrho$ there. We note that $ \hat{\Phi}^k$ is symmetric for any $ k$. Indeed: $ fT(g)(-\frac{k}{2})\,=\, t^{1/2}f g(-\frac{k}{2})\,=\,T(f)g(-\frac{k}{2})\ $ due to $ T=\frac{q^{2x+k/2}\!-q^{-k/2}}{q^{2x}-1}\,s- \frac{q^{k/2}\!-q^{-k/2}}{q^{2x}-1}$ and $ \ (q^{2x+k/2}-q^{-k/2})(x\mapsto -k/2)=0$.

Main Theorem 6.2 For $ F=fT(g)\in {\R}[X^{\pm 2}]$ and for any $ \Re k< 0$: $ G(k)\{f,g\}_\varkappa^\varrho=\Phi_0^k$ $ +\,\mu^\bullet(-k/2)\sum_{\tilde{k}\in \tilde{K}}\, A(\tilde{k})\,\left(\mu^\bullet(\tilde{k})/\mu^\bullet(-\frac{k}{2})\right) \,F(\tilde{k})$, for $ {\tilde{K}=\{n_\#,|n|\le m\}=\{-k/2\}\cup\{\pm\frac{k+j}{2},\, 1\!\le\!j\le m\},\ \,m\equal[\Re(-k)],}$ where $ [\cdot]\!=$ integer part, and $ \mu^\bullet(\pm\frac{k+j}{2})/\mu^\bullet(-\frac{k}{2})= t^{-j_\pm} \prod_{i=1}^{j_\pm}\frac{1\!-\!t^2q^i}{1\!-\!q^i}$ for $ j_+\!=\!j\!-\!1, j_-\!=\!j$. Generally, if $ F\in{\R}[X^{\pm 1}]$ ( not in $ {\R}[X^{\pm 2}]$ as above ), the poles of $ \mu$ are given by the relations $ q^{-\frac{1}{2}} X \in$ $ \pm\,$ $ q^{\Z_+/2}\,t^{\frac{1}{2}} \ni X^{-1}$, and the summation must be "doubled" accordingly. ⬜

Here we count "jumps" through the walls $ \Re k\!=\!-j\!\in\! -\Z_+$. The duplication of the summation for $ F\in{\R}[X^{\pm 1}]$ corresponds to the passage from affine Weyl group $ \tW$ to its extension $ \hWmm$ by $ \Pi=\Z_2$ in the $ p$-adic limit $ q\to 0, X\mapsto Y$ (discussed below).

Importantly, here and for any root systems only the total sum is an $ {\HH}$-invariant form. The partial sums with respect to the dimensions of the integration domains are symmetric and even $ \h_X$-invariant, but they are not $ {\HH}$-invariant.

The program is to $ (a)$ find (explicitly) these subtori for any root systems, $ (b)$ calculate the corresponding $ C$-coefficients (the $ q$-deformation of the AHA Plancherel measure), and finally $ (c)$ perform the $ p$-adic limit ($ q\to 0$), which steps are non-trivial even in type $ A$.

For the AHA $ \h$ of type $ A_1$, we set $ s=s_1, \om=\om_1, \pi=s\om$. Let \begin{eqnarray*} {\psi_n\!\equal\! t^{-\frac{|n|}{2}}T_{n\om}\p_+,\, \p_+\!=\!(1+t^{1/2}T)/(1\!+\!t) \for n\in \Z.} \end{eqnarray*} One can naturally consider them as polynomials in terms of $ {Y\equal T_{\om}=\pi T}$; then they become the Matsumoto spherical functions . This identification is based on the analysis by [Opdam2003] and the author: [Cherednik and Ostrik2003]; see also Cherednik ([Cherednik2006], Section 2.11.2), [Ion2006], [Cherednik and Ma2013]. Accordingly, the Satake–Macdonald $ p$- adic spherical functions become $ \,\p_+\psi_n (n\ge 0)$.

The corresponding version of the Main Theorem (compatible with the $ p$-adic limit) is as follows. The Gaussian collapses and we must omit it and use the integration over the period instead of the imaginary integration. We continue using the notations $ j_{\pm}\!=\! \{j-1,j\}, t=q^k$.

Let us summarize the main elements and steps of the construction we propose. The ingredients are as follows.

1. (a) Shapovalov anti-involution $ \varkappa$ of $ {\HH}$ (with respect to the subalgebra $ \y={\R}[Y_b]$), i.e. such that $ \{\varkappa(Y_a) T_w Y_b\}$ form a (PBW) basis of $ {\HH}$;
2. (b) the corresponding coinvariant : $ {\varrho:\HH\to {\R}}$ satisfying $ \varrho(\varkappa(H))=\varrho(H)$ (for any character of $ \y$ and $ \varrho$ on $ \H$ s.t. $ \varrho(T_w-T_{w^{-1}})\!=\!0$);
3. (c) the corresponding Shapovalov form $ {\{ f,g\}\equal}$ $ \varrho(A^\varkappa\, B)$ for $ {A,B\in \HH}$, satisfying $ \{ 1,1\}=1$ and analytic for any $ k$.

The $ W$-spinors are simply collections $ \{f_w,w\in W\}$ of elements $ f_w\in A$ with a natural action of $ W$ on the indices. If $ A$ (an algebra or a sheaf of algebras) has its own (inner) action of $ W$ and $ f_{w}=w^{-1}(f_{id})$, they are called principle spinors . Geometrically, hyperspinors are $ \C W$-valued functions on any manifolds, which is especially interesting for those with an action of $ W$. The technigue of spinors can be seen as a direct generalization of supermathematics , which is the case of the root system $ A_1$, from $ W=\S_2$ to arbitrary Weyl groups.

For instance, Laurent polynomials with the coefficients in the group algebra $ \C W$ are considered instead of $ \x$, the integration is defined upon the projection $ W\ni w\mapsto 1$ (a counterpart of taking the even part of a super-function), and so on and so forth. No "brand new" definitions are necessary here, but the theory quickly becomes involved.

The $ W$-spinors proved to be very useful for quite a few projects. One of the first instances was the author's proof in [1991] of the Cherednik–Matsuo theorem, an isomorphism between the AKZ and QMBP . An entirely algebraic version of this argument was presented in [Opdam1995]; also see [Cherednik2006]. This proof included the concept of the fundamental group for the configuration space associated with $ W$ or its affine analogs without fixing a starting point , à la Grothendieck. A certain system of cut-offs and the related complex hyperspinors can be used instead. The corresponding representations of the braid group becomes a $ 1$-cocycle on $ W$ (a much more algebraic object then the usual monodromy).

A convincing application of the technique of hyperspinors was the theory of non-symmetric $ q$-Whittaker functions. The Dunkl operators in the theory of Whittaker functions (which are non-symmetric as well as the corresponding Toda operators) simply cannot be defined without hyperspinors and the calculations with them require quite a mature level of the corresponding technique. See [Cherednik and Ma2013] and especially [Cherednik and Orr2015] (the case of arbitrary root systems). The Harish-Chandra-type decomposition formula for global nonsymmetric functions from [Cherednik2014] (for $ A_1$) is another important application; hyperspinors are essential here.

By the way, $ x^{2k}$ for complex $ k$, which is one of the key in the rational theory (see above), is a typical complex spinor , i.e. a collection of two (independent) branches of this function in the upper and lower half-planes. To give another (related) example, the Dunkl eigenvalue problem always has $ |W|$ independent spinor solutions; generally, only one of them is a function . In the case of $ {\HH''}$ for $ A_1$ (above), both fundamental spinor solutions for singular $ k=-1/2-m,\, m\in \Z_+$ are functions . See [Cherednik and Ma2013] for some details.

A natural question is, do we have hypersymmetric physics theories for any Weyl groups $ W$, say "$ W$-hypersymmetric Yang-Mills theory"? Jantzen filtration. It is generally a filtration of the polynomial representation of $ \x$ in terms of AHA modules , not DAHA modules, for $ \Re k< 0$. The top module is the quotient of $ \x$ by the radical of the sum of integral terms for the smallest residual subtori (points in many cases). Then we restrict the remaining sum to this radical and continue by induction with respect to the dimension of the (remaining) subtori.

Sometimes certain sums for residual subtori of dimensions smaller than $ n$ are DAHA-invariant; then $ \x$ is reducible. We expect that the reducibility of $ \x$ always can be seen this way, which includes the degenerations of DAHA. For $ A_n$, the corresponding Jantzen filtration provides the whole decomposition of $ \x$ in terms of irreducible DAHA modules, the so-called Kasatani decomposition ([Enomoto2009], [Etingof and Stoica2009]). Generally, the corresponding quotients can be DAHA-reducible.

For instance, the bottom module of the Jantzen filtration has the inner product that is (the restriction of) the integration over the whole $ i{\R}^n$. This provides some a priori way to analyze its signature (positivity), which is of obvious interest. The bottom DAHA submodule of $ \x$ was defined algebraically (without the Jantzen filtration) in [Cherednik2009a]. Indeed, it appeared semisimple under certain technical restrictions. For $ A_n$, this is related to the so-called wheel conditions .

Let us discuss a bit the rational case. The form $ \{f,g\}_\varkappa^\varrho$ for $ {\HH''}$ can be expected to have a presentations in terms of integrals over the $ x$-domains with $ \Re x$ in the boundary of a tube neighborhood of the resolution of the cross $ \prod_{\al\in R_+} (x,\al)=0$ over $ {\R}$. The simplest example is the integration over $ \pm i\ep +{\R}$ for $ A_1$. This resolution (presentation of the cross as a divisor with normal crossings) is due to the author (Publ. of RIMS, 1991), de Concini–Procesi, and Beilinson–Ginzburg.

This can be used to study the bottom module of the polynomial representation for singular $ k_o=-\frac{s}{d_i}$, assuming that it is well-defined and $ {\HH''}$-invariant. When $ s\!=\!1$ (not for any $ s$), it can be proved unitary in some interesting cases; see Etingof et al. in the case of $ A_n$ ([Etingof and Stoica2009]). The restriction of the initial (full) integration over $ {\R}^n$ provides a natural approach to this phenomenon. See Sect. 4 in the case of $ A_1$.

The DAHA-decomposition of the polynomial or other modules is a natural application of the integral formulas for DAHA-invariant forms, but we think that knowing such formulas is necessary for the DAHA harmonic analysis even if the corresponding modules are irreducible.