Universal nondegenerate affine Hecke ordered monomials form a basis

Ordered spanning without localization. For every $f\in P$ the presentation relations give $T_{i}f=s_i(f)T_i+c_{i,i+1}(f-s_i(f))$, where $c_{i,i+1}=(z-1)X_{i+1}\mathrm{/}(X_{i+1}-X_i)$. The correction is in $P$: $f-s_i(f)$ is divisible by $X_{i+1}-X_i$, first for ordinary monomials by the difference-of-powers identity, and then for Laurent monomials after multiplying by a symmetric Laurent monomial unit. For $f=X_i$ this is the Bernstein relation, for $f=X_{i+1}$ it is its conjugate using the quadratic relation, and the general formula follows by addition and the product rule $fg-s_i(fg)=(f-s_i(f))g+s_i(f)(g-s_i(g))$, including inverse generators by applying that rule to $f{f}^{-1}=1$. Moving a polynomial across a $T$ reduces the number of misplaced polynomial factors, and correction terms reduce the number of $T$ factors; the finite Hecke relations then span the remaining words by $T_w$. Thus $H_n^{\mathrm{a}\mathrm{f}\mathrm{f}}$ is spanned over $P$ by the $T_w$, and the presentation anti-involution fixing $X_i,T_i$ gives right ordered spanning too.

A representation of the unlocalized presentation. Form $K⋊S_n$ with multiplication $(f\sigma )(g\tau )=f\sigma (g)\sigma \tau$. For distinct indices set $c_{ij}=(z-1)X_j\mathrm{/}(X_j-X_i)$ and $d_{ij}=(X_j-z{X}_i)\mathrm{/}(X_j-X_i)=z-c_{ij}$, and assign $T_i\mapsto c_{i,i+1}+d_{i,i+1}{s}_i$, $X_j\mapsto X_j$. The rational identities $c_{ij}+c_{ji}=z-1$ and $d_{ij}{d}_{ji}=z+(z-1)c_{ij}-c_{ij}^2$ give the quadratic relation. For three distinct indices, expansion of the two braid products in their six permutation coefficients agrees: the longest coefficients are $d_{ij}{d}_{ik}{d}_{jk}$ in either order; the two length-two coefficients agree by commuting scalar factors; the length-one and identity coefficients agree using $c_{ij}{c}_{jk}=c_{ik}(c_{ij}+c_{jk}-(z-1))$ together with the preceding quadratic identities. These identities follow by multiplying by the common product of the differences and expanding the polynomial numerators. Distant relations follow from disjoint variable pairs. Finally $T_{i}f-s_i(f)T_i=c_{i,i+1}(f-s_i(f))$, so the Bernstein and mixed relations hold. All denominators here are nonzero polynomials in independent variables in the domain $P$. This defines a homomorphism from the unlocalized presented algebra to $K⋊S_n$.

Independence over the base ring. For a reduced word $w=s_{i_1}\cdots s_{i_k}$, the coefficient of the group element $w$ in the image of $T_w$ is $D_w=d_{i_1,i_1+1}{s}_{i_1}(d_{i_2,i_2+1})\cdots (s_{i_1}\cdots s_{i_{k-1}})(d_{i_k,i_k+1})\mathrm{\ne }0.$ Every factor is a nonzero rational function. Every other term uses at most $k-1$ of the $k$ adjacent transpositions, so its permutation has minimum word length at most $k-1$, directly by the definition of length in [F3]. No Bruhat-order assertion is needed. In a finite relation ${\sum }_w{f}_w{T}_w=0$ with $f_w\in P$, select a label $w$ of maximal length with $f_w\mathrm{\ne }0$. Its group-element coefficient is $f_w{D}_w$ and receives no contribution from any other label of that maximal length or from shorter labels, so it is nonzero in $K$, a contradiction. Hence the $T_w$ are $P$-linearly independent and the representation is injective. Since the Laurent monomials $X^a$ are an $R$-basis of $P$, this proves that $X^a{T}_w$ are an $R$-basis; the anti-involution proves the right basis. Tensoring this free basis with any commutative $R$-algebra preserves it, and tensoring the generators-and-relations presentation gives exactly the same affine presentation over that algebra. As the unused $U$ parameters do not enter the relations, any commutative coefficient ring with unit $z$ is obtained by sending every $U_a$ to $1$. Thus the asserted arbitrary base change is valid, without inverting differences in the base-changed algebra.

Localization and intertwiners. Let $S$ be the multiplicative set generated by all $X_i-X_j$ with $i\mathrm{\ne }j$. It is invariant under $S_n$. The mixed formula of step 1.1 extends to $S^{-1}P$ by the same product rule and by solving the formula for inverses of elements of $S$; the correction remains in $S^{-1}P$. Consequently adjoining these inverses gives the algebra with free left coefficient module ${⨁}_w(S^{-1}P)T_w$ and the same multiplication rules. This realizes the localization: the inverse formulas make it universal among homomorphisms inverting $S$, and its injective representation in $K⋊S_n$ follows from the identical maximal-length argument of step 1.3. In it put ${\mathrm{\Phi }}_i=T_i-c_{i,i+1}$; the mixed formula gives ${\mathrm{\Phi }}_{i}f=s_i(f){\mathrm{\Phi }}_i$ for all $f\in S^{-1}P$. Products ${\mathrm{\Phi }}_w$ along reduced words have unitriangular expansions in the $T_v$, because selecting every $T$ gives $T_w$ and selecting any scalar correction leaves strictly shorter words; thus they too form a basis over $S^{-1}P$. No parameter difference $U_a-U_b$, quantum factorial or Gram determinant is inverted.

Polynomial multiplication is injective over the universal domain. If $0\mathrm{\ne }f\in P$ and $h={\sum }_w{f}_w{T}_w$, then $fh=0$ gives $f{f}_w=0$ for each $w$ by step 1.3. Since $P$ is a domain, every $f_w=0$, so $h=0$. The right-basis argument gives the analogous right multiplication result. This injectivity assertion is over the universal domain, as in Statement (2), and does not assert injectivity of nonzero zero divisors after arbitrary base change.

Subword support in the localization. Expanding a reduced product $T_w={\prod }_t({\mathrm{\Phi }}_{i_t}+c_{i_t,i_t+1})$ and moving coefficients left using step 2.1 gives terms supported on products of subwords of the reduced word. Every coefficient is a sum of products of permuted $c$ functions and hence has only variable-difference denominators. Equivalently, in the crossed product ${\mathrm{\Phi }}_i=d_{i,i+1}{s}_i$, so moving a localized polynomial $f$ through such a term replaces it by the corresponding subword permutation applied to $f$. This proves the subword expansion used by consumers and the asserted intertwining throughout the embedded localization.

Conclusion. Step 1.3 proves both ordered bases over $R$ and after arbitrary base change with unit $z$; step 2.2 proves the universal-domain injectivity; and steps 2.1 and 3.1 prove the localized embedding and intertwining identity. For $n=0$ there are no polynomial or Hecke generators, $S_0$ consists of the identity, the lone ordered monomial is $1$, and $H_0^{\mathrm{a}\mathrm{f}\mathrm{f}}=R$; all localization and intertwining clauses are then vacuous. No cyclotomic basis, trace form, primitive-content assertion or Brundan–Kleshchev result is used.