Baghdad ? The elected Iraqi national assembly meets under heavy security in the so-called Green Zone, a huge chunk of the capital sealed off from the public by layers of tall concrete barriers and endless guards. To get inside the zone, a visitor goes through at least five security checks, with women guards, some veiled, frisking females to ensure they aren’t wearing explosives.

Inside the gloomy convention center, a fascinating mix of assembly members mills about: men in suits or open shirts, sheikhs in turbans, tribal figures in long cloaks, and women in day dress, or more likely, in heavy black abayas that look like nuns’ habits from the 1950s.

But the most interesting – and hopeful – political developments in Baghdad are going on behind the scenes.

Sunni Arabs, the Iraqi minority who held power under Saddam, make up the bulk of the insurgency. They boycotted January elections. But Sunnis now want to get into the political process – meaning the drafting of a new constitution and the next parliamentary elections slated for December.

The $64,000 question in Baghdad is whether the newfound Sunni political interest will undercut the insurgency – whether Sunni disaffection can be channeled into politics rather than guns.

“Whether we like it or not, significant parts of the Sunni community feel alienated,” I was told by Barham Salih, the Iraqi minister of planning, who is Kurdish. “You have to hit the insurgents hard, but without a political strategy to include Sunnis, this problem will continue.”

How to implement such a strategy has been controversial.

There is only a smattering of Sunnis in the national assembly, because of the boycott. The 55-person committee put together to draft a new constitution contained only two elected Sunnis.

Shiite leaders – who command a majority in the assembly – recognize the problem. Humam Hamoudi, an imposing, grey-bearded Shiite scholar in white turban, grey cloak, and black overcloak, chairs the constitutional committee. He told me: “We want and need Arab Sunnis to participate in the government and in the drafting of the constitution.”

The Shiites agreed to let Sunnis have six ministries, but Sunnis claim the names they suggested for the posts were ignored. Shiites argue back that they can’t figure out who speaks for the Sunnis. Saddam Hussein was the paramount Sunni leader, and he dispensed with competition.

Post-Saddam, Sunnis have no obvious leaders and have been divided into fragments. They have no paramount religious figure as the Shiites do in Ayatollah Ali al-Sistani.

But in recent weeks, prominent Sunnis have begun coalescing around the idea of participating in the political system. They agreed on a list of 15 unelected Sunnis who will offer advice on drafting the constitution – despite threats to their lives.

One of those, Salman al-Jumaili, a political science professor at Baghdad University, told me: “Realistic Sunnis think they must enter into the political process even under occupation. Everything indicates it’s not only force but a political process that is necessary to achieve our goals.”

This week, an umbrella group of religious and political organizations – the General Sunni Conference – announced Sunnis will fully participate in future elections. Sunni clerics may even urge worshippers to vote, as Ayatollah Sistani urged Shiites.

Some Shiite officials worry Sunnis will enter the constitutional process intending to disrupt it. Others suspect an IRA-style strategy of talk and fight. Sunnis will take part in politics while insurgents fight on, providing more leverage for Sunni demands. These include rehabilitation of senior Baathists and withdrawal of U.S. troops.

But overall, Shiite leaders insist they would rather have Sunnis inside than out of the process. There are efforts under way to set up goodwill committees to bridge huge political gaps and investigate the explosive issue of sectarian killings.

Meantime, the most promising idea for bringing more Sunnis into the system is to change the electoral law so members of the assembly are elected from provincial districts, instead of running on national party slates. This would mean that troubled Sunni provinces could choose local candidates. It would ensure that more Sunnis were elected to parliament.

There is no guarantee that Sunni politicians can or will undercut violent insurgents. But if they can offer their community a stake in the new system, they may ultimately isolate the hard core.