Admissible affine sl2 vacuum null polynomial

Fractional transport is an algebraic definition. For an original root generator f with locally nilpotent ad f, define on its Ore localization Theta_a(u)=sum_(j>=0) binom(a,j) (ad f)^j(u) f^-j.

The sum is finite for each u. It is an automorphism: for nonnegative integral a it is conjugation by $f^a$; each fixed coefficient identity is polynomial in a, so multiplicativity holds for every complex a. The inverse is ${\mathrm{\Theta }}_{-a}$. This gives a precise meaning to $f^a$ u $f^{-a}$. On a highest vector of h-weight b the usual finite commutator identity extends polynomially to e f^a v = a(b-a+1) f^(a-1) v.

Thus exponent $a=b+1$ performs the dot-reflection singular transport. Along a nonintegral simple wall it is invertible transport in the extended localization; at the middle integral wall it is the ordinary nonzero singular-vector embedding.

Put $t=k+2=p\mathrm{/}q$ and $m=p-1=qt-1$. Write the original simple coroots as $h_0=K-h$, $h_1=h$, with negative generators $f_0=e(-1)$ and $f_1=f(0)$. The vacuum shifted labels are $(⟨\lambda +\widehat{\rho },h_0⟩,⟨\lambda +\widehat{\rho },h_1⟩)=(t-1,1)$. Reflection at node $i$ negates its label and adds twice that label to the other, since the affine Cartan off-diagonal entries are $-2$. Alternating nodes $0,1,0,\dots$ therefore gives the successive reflected-node labels $a_j=(j+1)t-1$: $a_0=t-1$, $a_1=2t-1$, and $a_{j+1}=2a_j-a_{j-1}$. In the $2q-1$ steps only $a_{q-1}=qt-1=m$ is integral, because $\gcd (p,q)=1$. The first $q-1$ reflections transport ${\beta }_0=q\delta -\alpha$ to the middle simple root: iterating $s_i({\alpha }_j)={\alpha }_j+2{\alpha }_i$ for $i\mathrm{\ne }j$ verifies this (the middle root is ${\alpha }_0$ for odd $q$ and ${\alpha }_1$ for even $q$). Thus these outer steps are nonintegral equivalences and the middle step is the ordinary exponent-$m$ singular embedding. In operator order, rightmost factor first, the resulting product is $A=e(-1{)}^{m+(q-1)t}f(0{)}^{m+(q-2)t}e(-1{)}^{m+(q-3)t}\cdots f(0{)}^{m-(q-2)t}e(-1{)}^{m-(q-1)t}$: its exponents read from right to left are exactly $a_0,\dots ,a_{2q-2}$. This derives the gallery and every exponent from the affine Cartan relations, without an undefined $F_2$ premise.

The product has $2q-1$ factors; for $q=1$ it is the ordinary vector $e(-1{)}^{p-1}{v}_{\lambda }$. For $q>1$ its polynomiality requires the following local descent argument. All localizations in this argument invert just one original negative simple generator at a time. The next localization is performed only after the preceding expression has been proved polynomial.

Let $M(\nu )$ be a Verma, $f=f_i$, $a=\nu (h_i)\mathrm{\notin }\mathbb{Z}$, and $M(\nu {)}_f$ its Ore localization. For $c\in \mathbb{C}$ define the symbol module $f^{c}M(\nu {)}_f$ by $u(f^{c}v)=f^c{\mathrm{\Theta }}_{-c}(u)v$. This is an actual module since ${\mathrm{\Theta }}_{-c}$ is an automorphism; integral shifts of $c$ are identified by the invertible action of $f$. For $c=a+1$ the vector $w=f^c{v}_{\nu }$ is highest of weight $s_i\cdot \nu$: $e_{i}w=0$ by step 2.1, and $[e_j,f_i]=0$ for $j\mathrm{\ne }i$. It generates an injective copy of $M(s_i\cdot \nu )$. Indeed PBW identifies $M(\nu {)}_f$ with $U({\mathfrak{n}}_{-}{)}_f{v}_{\nu }$, and $Pw=f^c{\mathrm{\Theta }}_{-c}(P)v_{\nu }$ is nonzero for every nonzero $P\in U({\mathfrak{n}}_{-})$. Its $f$-localization is the entire symbol module, by the same automorphism and PBW identification.

No new vector killed by $e_i$ occurs in $M(s_i\cdot \nu {)}_f\mathrm{/}M(s_i\cdot \nu )$. To see this, use [F1, Proof 29.1–30.1]: in each full-Cartan coset modulo $\mathbb{Z}{\alpha }_i$, the Verma is a finite direct sum of highest ${\mathfrak{s}\mathfrak{l}}_2^i$ strings, all with nonintegral highest weights $d$. Their decomposition uses only the nonzero coefficients $r!{\prod }_{j=0}^{r-1}(d-j)$ and finite local nilpotence of $e_i$. Localization and quotient commute with this direct sum. Each quotient string has basis $f^{-r}{v}_d$, $r\ge 1$, and $e_i{f}^{-r}{v}_d=-r(d+r+1)f^{-r-1}{v}_d$. Every coefficient is nonzero, so $e_i$ is injective on the quotient. Hence every singular vector in this localized Verma already belongs to the unlocalized Verma.

Suppose $P{v}_{\nu }$ is a nonzero singular vector of weight $\mu$, put $b=\mu (h_i)$, and assume $b-a\in \mathbb{Z}$. In the symbol module of step 6.1 the vector $f^{b+1}P{v}_{\nu }$ is nonzero and singular, by the same ${\mathfrak{s}\mathfrak{l}}_2$ identity and $[e_j,f_i]=0$ for $j\mathrm{\ne }i$. Relative to $w=f^{a+1}{v}_{\nu }$ it is the finite localized expression $Qw$, where $Q=f^{b+1}P{f}^{-a-1}={\mathrm{\Theta }}_{b+1}(P)f^{b-a}\in U({\mathfrak{n}}_{-}{)}_f$. Step 7.1 puts $Qw$ inside $M(s_i\cdot \nu )$, so injective PBW localization forces $Q\in U({\mathfrak{n}}_{-})$. It is therefore a nonzero polynomial singular vector of weight $s_i\cdot \mu$. This proves polynomial descent, rather than identifying arbitrary localized Hom spaces.

Start at the middle ordinary embedding $f_{i_{q-1}}^m$, where $i_j=0$ for even $j$ and $i_j=1$ for odd $j$, and reverse the first $q-1$ reflections of step 4.1. At each reverse step the target highest label is nonintegral, and the source differs by an integral root-lattice weight, so step 8.1 applies. For the reverse step at node $i_j$, the shifted target label is $-a_j$ and the shifted source label is $a_{2q-2-j}$, as follows by the same two-label reflection recurrence of step 4.1. Thus the recursion is $P_j=f_{i_j}^{a_{2q-2-j}}{P}_{j+1}{f}_{i_j}^{a_j}$. Every $P_j$ is polynomial and nonzero by step 8.1. Expanding this finite nested recursion gives exactly the displayed alternating product $A$. It is the actual singular embedding for ${\beta }_0=q\delta -\alpha$, not just a localized candidate. Its nonzero vacuum image follows from the degree-one ordinary BGG generators: the ${\beta }_0$ label and the finite simple label are distinct covers, so strong linkage excludes its highest simple factor from the finite-simple Verma submodule. After the finite-simple quotient its image therefore generates the actual maximal vacuum ideal by [F1].

Evaluate the product, without a projection citation. Let P1 be evaluation on the negative affine algebra: x(-r) maps to x for $r\ge 1$ and f(0) maps to f. Negative-loop commutators have no central term, so P1 is a Lie algebra map and extends to enveloping algebras and the intermediate Ore transports. Put H_a=fe-a h-a(a+1).

Direct use of $[e,f]=h,[h,e]=2e$ gives e^b H_a=H_(a-b)e^b, f^b H_a=H_(a+b)f^b, e^m f^m=product_(r=1)^m H_-r, f^m e^m=product_(r=0)^(m-1) H_r.

The shift identities extend to complex b by Theta. The $H_a$ commute because each is a polynomial in the commuting operators fe and h.

Induct on the number Q of e factors in the alternating palindrome with step t. Its evaluation is E_Q(m)=product_(r=1)^m product_(s=1)^(Q-1) H_(-r-st) e^m.

The companion palindrome beginning with f is F_Q(m)=product_(r=0)^(m-1) product_(s=1)^(Q-1) H_(r+st) f^m.

For $Q=1$ these are identities. To prove the e formula from the f formula, write $E_Q=e^{m+(Q-1)t}$ $F_{Q-1}$ $e^{m-(Q-1)t}$. Move its left e power past the H factors. The indices become r-m+(s-Q+1)t, which reindex as -r'-s't for $1\le r^{\mathrm{\prime }}\le m,1\le s^{\mathrm{\prime }}\le Q-2$. The remaining $e^{m+(Q-1)t}$ $f^m$ $e^{m-(Q-1)t}$ contributes precisely the missing $s^{\mathrm{\prime }}=Q-1$ factors, by $e^m$ $f^m$ and the shift identity. The f formula follows by the reversed calculation. This proves by simultaneous induction the required exact projection P1(A)=product_(r=1)^m product_(s=1)^(q-1) H_(-r-st) e^m.

In particular the evaluated polynomial is nonzero. No cancellation or generic-level assumption has replaced this calculation.

Extract the highest-top annihilation polynomial. Suppose the highest affine module L(k,j) is an actual module for the simple vacuum quotient. Let u be its horizontal highest vector, h $u=j$ u. The vacuum singular state A1 acts by the zero field. Use the skew intertwiner $I(v,z)a=\exp (zL(-1))Y(a,-z)v$ and pair with the functional u' on the HIGHEST h-weight line of the grade-zero horizontal module, extended by zero on all other weights and grades. For x(-r), $r\ge 1$, the current commutator has zero output term after this projection, since it raises loop grade; its remaining term is $-z^{-r}$ x on v. For the remaining negative-Borel generator f(0), its output term also vanishes because the functional is on the highest h-weight line and f lowers every horizontal weight. Applying these identities one factor at a time reverses their order and gives <u', I(v,z) a1> = z^(loop-degree a) <u', I(sigma P1(a)v,z)1>,

where sigma is the antipode a↦-a of U(sl2). Terms with nonzero final loop grade disappear against the grade-zero functional; the current commutator supplies exactly the evaluation P1 and the antipode, with no analytic continuation. Translation in I(v,z)1 contributes only its constant term to this pairing. Thus sigma P1(A) kills the horizontal top in the required matrix coefficients.

Since $\sigma (H_a)=H_{-a-1}$, the projection formula gives sigma P1(A)=(-1)^m product_(r=1)^m product_(s=1)^(q-1) H_(-p+r+st) e^m.

Apply this to $f^m$ u. On u, $H_a$ acts by -a(j+a+1), and $e^m$ $f^m$ acts by $produc{t}_{i=1}^m$ i(j-i+1). The remaining scalar factors p-r-st are nonzero: equality would imply q divides s, impossible for $1\le s\le q-1$. Removing these nonzero constants leaves exactly product_(r=0)^(p-2) product_(s=0)^(q-1) (j-r+st)=0.

Therefore $j=r-st$ with $0\le r\le p-2$ and $0\le s\le q-1$. Conversely all these highest modules descend by the G-integrable construction followed by the nonintegral-wall/length-zero transports, whose explicit module proof is part of the shared affine construction. Only necessity was needed from the null-field computation itself.

These roots are exactly the rank-one admissible weights, by direct arithmetic: the integral positive boundary roots are alpha+s delta and -alpha+(q-s)delta (using alpha itself if $s=0$); their shifted pairings are r+1 and p-r-1, both positive. The period is q, so their rational span is full. If j is integral, coprimality forces $s=0$, giving $0\le j\le p-2$. No Kac–Wakimoto classification proposition is being used for this rank-one conclusion.

The same calculation is an OPERATOR identity on an arbitrary generalized horizontal highest space U with $eU=0$, not only a scalar equation: replace j by the endomorphism h in the displayed products and pair against every functional on U. The scalar constants removed above are still nonzero. Thus $f(h)U=0$. Its roots are pairwise distinct: equality $r-st=r^{\mathrm{\prime }}-s^{\mathrm{\prime }}t$ implies $q(r-r^{\mathrm{\prime }})=p(s-s^{\mathrm{\prime }})$, hence $s=s^{\mathrm{\prime }}$ from the range $0\le s,s^{\mathrm{\prime }}<q$ and coprimality, then $r=r^{\mathrm{\prime }}$. Consequently h is semisimple on every finite-dimensional such U. This is the specific generalized-highest-space consequence available for the later principal-W self-extension argument; it is not a generic reverse-Zhu criterion.