Cyclotomic Level Maps

• Cyclotomic level maps are context-dependent constructions that assign discrete levels to cyclotomic phenomena across Lie theory, stable homotopy, and arithmetic.
• In Lie theory, they classify nilpotent orbits and Weyl-group conjugacy classes by leveraging methods such as Bala–Carter data and compatibility with Lusztig’s and Yun’s maps.
• In topology and arithmetic, they underpin Frobenius-type comparisons and control quadratic extensions, unifying various mathematical areas through level extraction.

Searching arXiv for papers specifically on “cyclotomic level maps” and closely related usage. Cyclotomic level maps are constructions that attach “levels” to cyclotomic phenomena, but the phrase is used in several distinct mathematical settings. In recent Lie-theoretic usage, it denotes two maps with values in positive integers, one on nilpotent orbits and one on Weyl-group conjugacy classes, designed to be compatible with Lusztig’s map and Yun’s minimal reduction type map; these maps are then used to formulate a conjecture on associated varieties of simple affine vertex algebras at non-admissible integer levels (Shan et al., 12 Jul 2025). In stable homotopy theory, the same phrase refers to levelwise Frobenius-type maps in cyclotomic spectra or to higher-root-of-unity maps such as $B^n\mathbb{C}_{p^j}\to R^\times$ in higher cyclotomic extensions (Ravenel, 6 Jun 2026, Ravenel, 6 Jun 2026). In arithmetic field theory, it appears as the function $t_F:\mathbb N\to\mathbb N$, introduced to control primitive roots of unity defining quadratic cyclotomic extensions (Marques et al., 2022).

1. Lie-theoretic definition on nilpotent orbits and Weyl conjugacy classes

For a complex simple Lie algebra $\mathfrak g$, let $\mathcal N$ be its nilpotent cone and $\underline{\mathcal N}$ the set of adjoint nilpotent orbits. The nilpotent-orbit cyclotomic level map is

$\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$

If $\mathcal O\in \underline{\mathcal N}$ is written in Bala–Carter form

$\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$

where $L\subset G$ is a Levi subgroup and $\mathcal O_L$ is distinguished in $t_F:\mathbb N\to\mathbb N$0, one chooses an $t_F:\mathbb N\to\mathbb N$1-triple $t_F:\mathbb N\to\mathbb N$2 with $t_F:\mathbb N\to\mathbb N$3, lets $t_F:\mathbb N\to\mathbb N$4 be the largest $t_F:\mathbb N\to\mathbb N$5-weight occurring in $t_F:\mathbb N\to\mathbb N$6, and defines

$t_F:\mathbb N\to\mathbb N$7

Equivalently,

$t_F:\mathbb N\to\mathbb N$8

The construction is stated to be independent of the choices involved and to depend only on the Bala–Carter data of $t_F:\mathbb N\to\mathbb N$9 (Shan et al., 12 Jul 2025).

The second map is defined on Weyl-group conjugacy classes. For $\mathfrak g$0, let $\mathfrak g$1 be its characteristic polynomial in the reflection representation $\mathfrak g$2. Since $\mathfrak g$3 is defined over $\mathfrak g$4, $\mathfrak g$5 factors into cyclotomic polynomials $\mathfrak g$6. One defines

$\mathfrak g$7

which descends to

$\mathfrak g$8

The terminology “cyclotomic” comes from this factorization into cyclotomic polynomials (Shan et al., 12 Jul 2025).

A central feature of the theory is that the two maps are designed to be compatible: the paper proves compatibility with the minimal reduction type map, and in the simply-laced case they also match the nilpotent-orbit map through Lusztig/Yun’s correspondence (Shan et al., 12 Jul 2025).

2. Order structure and explicit formulas

The map $\mathfrak g$9 organizes nilpotent orbits by level. For each $\mathcal N$0 in the image of $\mathcal N$1, there is a unique maximal orbit $\mathcal N$2 among those with $\mathcal N$3, and

$\mathcal N$4

Moreover, if $\mathcal N$5, then

$\mathcal N$6

and if $\mathcal N$7, then

$\mathcal N$8

This gives a filtration of nilpotent orbits by cyclotomic level (Shan et al., 12 Jul 2025).

For the classical types, the values are given explicitly in terms of the partition parametrization of nilpotent orbits.

Type	Formula for $\mathcal N$9	Image of $\underline{\mathcal N}$0
$\underline{\mathcal N}$1	$\underline{\mathcal N}$2	$\underline{\mathcal N}$3
$\underline{\mathcal N}$4	$\underline{\mathcal N}$5 if $\underline{\mathcal N}$6; $\underline{\mathcal N}$7 if $\underline{\mathcal N}$8	$\underline{\mathcal N}$9
$\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$0	$\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$1	$\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$2
$\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$3	$\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$4 if $\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$5; $\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$6 if $\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$7	$\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$8

In type $\cl_n : \underline{\mathcal N} \to \mathbb Z_{\ge 1}.$9, if $\mathcal O\in \underline{\mathcal N}$0 corresponds to the partition $\mathcal O\in \underline{\mathcal N}$1, then $\mathcal O\in \underline{\mathcal N}$2. For $\mathcal O\in \underline{\mathcal N}$3, the maximal orbit $\mathcal O\in \underline{\mathcal N}$4 has partition

$\mathcal O\in \underline{\mathcal N}$5

where $\mathcal O\in \underline{\mathcal N}$6 is maximal with $\mathcal O\in \underline{\mathcal N}$7 and $\mathcal O\in \underline{\mathcal N}$8 (Shan et al., 12 Jul 2025).

For the exceptional types $\mathcal O\in \underline{\mathcal N}$9, the values are computed case by case using the atlas software, and the resulting figures list Bala–Carter labels, $\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$0-values, and the orbits $\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$1. The paper records, for example, that in $\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$2 the orbit $\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$3 has $\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$4 and is $\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$5 (Shan et al., 12 Jul 2025).

3. Compatibility with Lusztig’s map, Yun’s reduction type, and affine cells

Yun’s construction associates to a regular semisimple topologically nilpotent $\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$6 its reduction type $\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$7, the set of nilpotent orbits occurring as reductions modulo $\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$8 of $\mathcal O=\operatorname{Sat}_L^G(\mathcal O_L),$9-conjugates of $L\subset G$0. The minimal reduction type $L\subset G$1 is the set of minimal such orbits. Yun proves that for each conjugacy class $L\subset G$2, the intersection of $L\subset G$3 over the shallow stratum of type $L\subset G$4 is a singleton, yielding a map

$L\subset G$5

The main compatibility theorem states

$L\subset G$6

Equivalently, the triangle with vertices $L\subset G$7, $L\subset G$8, and $L\subset G$9 commutes (Shan et al., 12 Jul 2025).

A key proposition establishes that for any nilpotent orbit $\mathcal O_L$0,

$\mathcal O_L$1

where $\mathcal O_L$2 is the Kazhdan–Lusztig map from nilpotent orbits to Weyl conjugacy classes. Together with Yun’s identification $\mathcal O_L$3, this yields the compatibility of the two level maps (Shan et al., 12 Jul 2025).

The same paper places cyclotomic level maps in the theory of two-sided cells of affine Weyl groups. Lusztig constructed a bijection

$\mathcal O_L$4

where $\mathcal O_L$5 is the nilpotent cone of the Langlands dual Lie algebra. For $\mathcal O_L$6, let $\mathcal O_L$7 be the unique dominant affine weight in the $\mathcal O_L$8-orbit of $\mathcal O_L$9, let $t_F:\mathbb N\to\mathbb N$00, and let $t_F:\mathbb N\to\mathbb N$01 be the longest element in $t_F:\mathbb N\to\mathbb N$02. If $t_F:\mathbb N\to\mathbb N$03 is the largest value in the image of $t_F:\mathbb N\to\mathbb N$04 with $t_F:\mathbb N\to\mathbb N$05, then Lusztig’s bijection sends the two-sided cell of $t_F:\mathbb N\to\mathbb N$06 to $t_F:\mathbb N\to\mathbb N$07: $t_F:\mathbb N\to\mathbb N$08 The paper also proves

$t_F:\mathbb N\to\mathbb N$09

so the two-sided cell depends only on the cyclotomic level cutoff $t_F:\mathbb N\to\mathbb N$10 (Shan et al., 12 Jul 2025).

4. Associated varieties of simple affine vertex algebras

For a complex simple Lie algebra $t_F:\mathbb N\to\mathbb N$11, the simple affine vertex algebra $t_F:\mathbb N\to\mathbb N$12 is the simple quotient of the universal affine vertex algebra $t_F:\mathbb N\to\mathbb N$13 at level $t_F:\mathbb N\to\mathbb N$14. Its associated variety is

$t_F:\mathbb N\to\mathbb N$15

a conical $t_F:\mathbb N\to\mathbb N$16-stable Poisson subvariety of $t_F:\mathbb N\to\mathbb N$17. The known cases recorded in the paper are the following: if $t_F:\mathbb N\to\mathbb N$18 is nonnegative rational in the universal range, then $t_F:\mathbb N\to\mathbb N$19 and $t_F:\mathbb N\to\mathbb N$20; at the critical level $t_F:\mathbb N\to\mathbb N$21, one has $t_F:\mathbb N\to\mathbb N$22; and for admissible levels, $t_F:\mathbb N\to\mathbb N$23 is known to be the closure of an explicitly determined nilpotent orbit (Shan et al., 12 Jul 2025).

The proposed application of cyclotomic level maps concerns non-admissible integer levels. Assuming $t_F:\mathbb N\to\mathbb N$24 is simply laced, let

$t_F:\mathbb N\to\mathbb N$25

Let $t_F:\mathbb N\to\mathbb N$26 be the largest number in the image of the dual cyclotomic map $t_F:\mathbb N\to\mathbb N$27 such that $t_F:\mathbb N\to\mathbb N$28, and write

$t_F:\mathbb N\to\mathbb N$29

with $t_F:\mathbb N\to\mathbb N$30 distinguished in a Levi $t_F:\mathbb N\to\mathbb N$31. The conjecture is

$t_F:\mathbb N\to\mathbb N$32

Here $t_F:\mathbb N\to\mathbb N$33 is Barbasch–Vogan–Lusztig–Spaltenstein duality, and $t_F:\mathbb N\to\mathbb N$34 is the sheet attached to $t_F:\mathbb N\to\mathbb N$35, defined as the image of

$t_F:\mathbb N\to\mathbb N$36

The conjectural associated variety is therefore often a sheet closure rather than merely a nilpotent orbit closure (Shan et al., 12 Jul 2025).

The paper derives a quasi-lisse criterion from the conjecture. Since $t_F:\mathbb N\to\mathbb N$37 is conical, quasi-lisseness is equivalent here to $t_F:\mathbb N\to\mathbb N$38. The conjecture implies that $t_F:\mathbb N\to\mathbb N$39 is quasi-lisse if and only if $t_F:\mathbb N\to\mathbb N$40 is distinguished, in which case

$t_F:\mathbb N\to\mathbb N$41

Evidence is provided by comparison with known computations of Arakawa, Arakawa–Moreau, Arakawa–Futorny–Křížka, Jiang–Song, and Gorelik–Kac, by a detailed type $t_F:\mathbb N\to\mathbb N$42 example, and by separate checks of the regular case $t_F:\mathbb N\to\mathbb N$43, the subregular case, and the case $t_F:\mathbb N\to\mathbb N$44 (Shan et al., 12 Jul 2025).

5. Homotopy-theoretic meanings: levelwise Frobenius maps and higher roots of unity

In stable homotopy theory, cyclotomic spectra provide a different use of “cyclotomic level maps.” In the geometric or orthogonal-spectrum formulation, a cyclotomic spectrum is an orthogonal $t_F:\mathbb N\to\mathbb N$45-spectrum $t_F:\mathbb N\to\mathbb N$46 equipped with $t_F:\mathbb N\to\mathbb N$47-equivariant maps

$t_F:\mathbb N\to\mathbb N$48

for each prime $t_F:\mathbb N\to\mathbb N$49 and each $t_F:\mathbb N\to\mathbb N$50-representation $t_F:\mathbb N\to\mathbb N$51, satisfying conditions implying an isomorphism

$t_F:\mathbb N\to\mathbb N$52

These are the original levelwise maps. In the Nikolaus–Scholze formulation, the same structure is repackaged as a global cyclotomic structure map

$t_F:\mathbb N\to\mathbb N$53

For bounded-below spectra, the two definitions are stated to agree (Ravenel, 6 Jun 2026).

A Floer-theoretic realization appears in symplectic topology. For a compact symplectically atoroidal manifold $t_F:\mathbb N\to\mathbb N$54 with contact boundary and an equivariant trivialization of the polarization class, the paper constructs an object

$t_F:\mathbb N\to\mathbb N$55

in the $t_F:\mathbb N\to\mathbb N$56-category of genuine $t_F:\mathbb N\to\mathbb N$57-cyclotomic spectra with

$t_F:\mathbb N\to\mathbb N$58

The cyclotomic structure is expressed by the equivalence

$t_F:\mathbb N\to\mathbb N$59

modeled on the free-loop-space homeomorphism

$t_F:\mathbb N\to\mathbb N$60

In this setting, the level maps are a compatible family across the tower of $t_F:\mathbb N\to\mathbb N$61-spectra rather than a single endomorphism (Rezchikov, 2024).

A second homotopy-theoretic usage occurs in higher cyclotomic extensions of spectra. There a height $t_F:\mathbb N\to\mathbb N$62 $t_F:\mathbb N\to\mathbb N$63-th root of unity in a commutative ring spectrum $t_F:\mathbb N\to\mathbb N$64 is defined as a map

$t_F:\mathbb N\to\mathbb N$65

equivalently

$t_F:\mathbb N\to\mathbb N$66

These maps underlie the extensions

$t_F:\mathbb N\to\mathbb N$67

and lead to the cyclotomic completion functors determined by

$t_F:\mathbb N\to\mathbb N$68

The associated Bousfield classes satisfy

$t_F:\mathbb N\to\mathbb N$69

so cyclotomic completion interpolates between chromatic localization and telescopic localization (Ravenel, 6 Jun 2026).

6. Arithmetic level functions and related degree-$t_F:\mathbb N\to\mathbb N$70 trace phenomena

In the arithmetic theory of quadratic cyclotomic extensions, the cyclotomic level map is the function

$t_F:\mathbb N\to\mathbb N$71

defined prime-power-wise and then multiplicatively. For a prime $t_F:\mathbb N\to\mathbb N$72 and $t_F:\mathbb N\to\mathbb N$73, the paper defines $t_F:\mathbb N\to\mathbb N$74 by cases according to the value of $t_F:\mathbb N\to\mathbb N$75, and then sets

$t_F:\mathbb N\to\mathbb N$76

Its purpose is to encode the “effective part” of a root of unity that controls whether it defines a quadratic cyclotomic extension (Marques et al., 2022).

The same paper studies primitive $t_F:\mathbb N\to\mathbb N$77th roots of unity $t_F:\mathbb N\to\mathbb N$78 such that

$t_F:\mathbb N\to\mathbb N$79

If $t_F:\mathbb N\to\mathbb N$80, then the minimal polynomial of $t_F:\mathbb N\to\mathbb N$81 over $t_F:\mathbb N\to\mathbb N$82 is

$t_F:\mathbb N\to\mathbb N$83

for a unique $t_F:\mathbb N\to\mathbb N$84 with $t_F:\mathbb N\to\mathbb N$85, and

$t_F:\mathbb N\to\mathbb N$86

For odd primes $t_F:\mathbb N\to\mathbb N$87, if $t_F:\mathbb N\to\mathbb N$88, then

$t_F:\mathbb N\to\mathbb N$89

For powers of $t_F:\mathbb N\to\mathbb N$90, if $t_F:\mathbb N\to\mathbb N$91 and $t_F:\mathbb N\to\mathbb N$92, then

$t_F:\mathbb N\to\mathbb N$93

The paper also determines the maximal natural number $t_F:\mathbb N\to\mathbb N$94 such that $t_F:\mathbb N\to\mathbb N$95 defines a quadratic cyclotomic extension over $t_F:\mathbb N\to\mathbb N$96, with a uniform characterization covering both odd and even primes (Marques et al., 2022).

A further, different appearance of the phrase occurs in the $t_F:\mathbb N\to\mathbb N$97-completed cyclotomic trace in degree $t_F:\mathbb N\to\mathbb N$98. For a quasi-regular semiperfectoid $t_F:\mathbb N\to\mathbb N$99-algebra $\mathfrak g$00, the paper identifies

$\mathfrak g$01

describing this eigenspace as the key target of the degree-$\mathfrak g$02 cyclotomic trace. Under the canonical class from the Tate module of units, the composition

$\mathfrak g$03

is given by

$\mathfrak g$04

where $\mathfrak g$05 is the $\mathfrak g$06-deformation of the logarithm. The same paper proves that

$\mathfrak g$07

is a bijection for any quasi-regular semiperfectoid $\mathfrak g$08-algebra $\mathfrak g$09 (Anschütz et al., 2019).

Across these examples, the phrase does not denote a single universal construction. This suggests that “cyclotomic level map” functions as a context-dependent term whose common feature is the extraction of a discrete level from cyclotomic structure: a Bala–Carter depth on nilpotent orbits, a maximal cyclotomic factor in Weyl-group characteristic polynomials, a levelwise Frobenius comparison in cyclotomic spectra, a higher-root-of-unity map in chromatic homotopy theory, or an arithmetic reduction $\mathfrak g$10 governing quadratic cyclotomic extensions.